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Maximum principles and Liouville theorems for elliptic partialdifferential equations
Zhou, Chiping, Ph.D.
University of Hawaii, 1990
V·M·I300 N. Zeeb RdAnn Arbor, MI48106
MAXIMUM PRINCIPLES AND LIOUVILLE THIDREMS
FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF
THE UNIVERSITY OF HAWAI I IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN MATHEMATICS
AUGUST 1990
By
Chiping Zhou
Dissertation Committee:
Gerald N. Hile, Chairman
Thomas B. Hoover
Marvin E. Ortel
Xerxes R. Tata
Rui Zong Yeh
ACKNOWLEDGEMENTS
I wish to express my sincere thanks to my advisor, Professor
Gerald N. Hile, for his helpful suggestions and for his encouragement
in writing this dissertation. It has been a pleasure working under
the guidance of a leading researcher, who with his tremendous
expertise has kept my research running smoothly.
In addition, I would like to acknowledge all those who, directly
or indirectly, helped with my mathematical education, especially,
Chuanzhong Chen, Zongyi Hou, Mingzhong Li, Robert P. Gilbert, William
Lampe, Wayne Smith, and the members of my dissertation committee.
Finally, I wish to thank my wife Xiaoyu for her understanding and
support during my mathematical endeavours.
iii
iv
ABSTRACT
Part I : Maximum Principles for Elliptic Systems 0
Maximum principles and bounds of solutions for weakly coupled
second order elliptic systems,
Lu + Cu = 0 or f in D C IRn,
are established by using the idea of Liapunov's Second Method. Here
n a2L:= L a. o (x)ax ax +
10 JO -1 1 J 10 JO, -n aL bo(X)ax
i=l 1 i
is a second order real elliptic operator, and the mx1 vectors f, u,
and the mxm matrix C all have entries which are complex-valued
functions. It is shown that:
*If there exists a complex constant matrix B > ° such that C (x)B
+ BC(x) :;;; ° cor « -E < .0, E is a constant matrix) in D, then for all
C2(D)nC(D) solutions u of Lu + Cu = ° (or = f),
Ilullo,D:;;; K1l1ullo,aD (or Ilullo,D:;;; K11lullo,aD + K2I1fllo,D).
(P 1/2 2K
1Here K = __!!!..J and K = -- , where PI and f3 are the smallest1 PI ) 2 J.1.1 m
and biggest eigenvalues of B, respectively, and J.1.1
is the smallest
eigenvalue of EB-1.
Part II Schauder Estimates and Existence Theory for Entire Solutions
of Linear Fourth Order Elliptic Equations 0
Solutions in IRn (n ~ 2) of the linear, fourth order, variable
coefficient, elliptic equation,
and the
n 84p}: aiJ'kl ax ax ax ax +
i,j,k,l=l i j k I
n 82p n f!!!..-+ }: c . . ax ax + }: d . ax + e,p = f,
i,j=l 1J i j i=l 1 i
related homogeneous equation are investigated. An entire
v
solution is a solution of the equation defined in all 'IRn ,
Schauder-type a priori estimates are developed for entire solutions
with prescribed behavior at infinity. These estimates, of perhaps
independent interest themselves, lead to an existence and uniqueness
theory for entire solutions, with certain behavior required at
infinity, when the operator L can be separated as L = L2L 1
" Here L1
and L2
are two second order elliptic operators.
Under some suitable conditions that guarantee the operator L
approaches the biharmonic operator a2 near infinity at a certain
rate, it is shown that there exists a unique entire solution in IRn
(n ~ 2) of Lr/J = f (f(x) = O(lx\-4-E) near infinity), such that
4-n~(x) - L 1 .Dsr(x) ~ 0 as x ~ 00,ss=o
for some constant number 10
, constant vector ~l and constant symmtric
matrix 12
, Here r is a fundamental solution of the biharmonic
equation, Moreover, a one-to-one correspondence between the entire
4-nsolutions of Lr/J = f, with ,p(x) - }: 1 .Dsr(x) = O(lxl m) nearss=o
infinity, and the biharmonic polynormials of degree no greater than m
is established. When f =0, the result is an extension of a
Liouville-type theorem.
TABLE OF CONTENTS
Acknowledgements
Abstract
Part I. Maximum Principles for Elliptic Systems
1. Introduction
2. Notation and a Liapunov Stablity Theorem
3. Maximum Principles .....
4. Uniqueness Theorems for Some Boundary Value Problems
5. Estimate of K
6. Bounds for Solutions of the Nonhomogeneous Systems
7. Applications to Higher Order Equations ..
References for Part I . . .
iii
iv
1
2
4
6
13
17
24
28
32
vi
Part II. Schauder Estimates and Existence Theory for Entire
Solutions of Linear Fourth Order Elliptic Equations 35
1. Introduction. 36
2. Norms and a Schauder Estimate for Entire Solutions 38
3. Bounds for Entire Solutions of the Nonhomogeneous
Biharmonic Equation . . . . . . . . . . . 47
4. An A Priori Bound for the Fourth Order Elliptic Equation 63
5. Existence of Solutions
References for Part II
83
94
1. INTRODUCTION
*Positive definite solutions B of the matrix equation C B + BC =
-E (E > 0) have been successfully used to construct Liapunov
functions, and then to prove the stability of some ordinary
2
differential systems dudx = Cu (cf: [11], [24]). This method usually
is called Liapunov's Second Method. In 1974, Chow and Dunninger [2]
applied this method to the study of n-metaharmonic functions, and
obtained a generalized maximum principle for some classes of
n-metahamonic functions.
In this paper, we transfer the idea of Liapunov's second method
to the study of weakly coupled second-order elliptic systems
Here
Lu + Cu = 0 or f in D C IRn .
a2 8L == a .. (x)ax ax + a. (x)-a '
IJ ., 1 X.1 J 1
a .. = a ..IJ J 1
Tis a second-order real elliptic operator, and f = (f1
, .,. ,fm) ,T'
u = (u1
' .. , ,urn) ,and the mxm matrix C all have entries which are
complex-valued functions.
We establish the following generalized maximum principle for
certain classes of the homogeneous system (Theorem 3.1);
If there exists a complex crnstant matrix B > 0 such that
C*(x)B + BC(x) ~ 0 in D, then for all C2 (D) nC(D) solutions u of
Lu + CU = 0 ,
lI ull o,D ~ K lIullo,8D .
Here K ((3(3m ) 1/2 , where (3 and (3 are the smallest and biggest
1 m1
eigenvalues of B, respectively.
We also find a simple sufficient condition for the classical maximum
principle (K = 1 in the above inequality) holding; this condition is
*C (x) + C(x) ~ 0 in D (Theorem 3.3). These results extend the result
of Winter and Wong [23] for C being negative semidefinite to a more
general form of C. Generalized maximum principles for weakly coupled
3
in D C IRn
second-order elliptic systems have also been obtained by Dow [3], Hile
and Protter [8], Szeptycki [21] and Wasowski [22] under different
conditions on the coefficients.
We further show how our maximum principles may be used to prove
the uniqueness of various boundary value problems of some classes of
elliptic systems over bounded or unbounded domain D C ~n. By using a
recent result of Hile and Yeh [10], we even obtain uniqueness for a
boundary value problem with an exceptional boundary set r c an with
the Hausdorff dimension of r less than n - 1.
An estimate of the best possible K in our maximum principle
inequality is given when C is a 2 by 2 real matrix. The condition for
the classical maximum principle (K = 1) holding can be written as
Re{a} ~ 0, Re{d} ~ 0, Ib + cl 2~ 4Re{a}Re{d}
when C = [:~~~ ~~~~] is any 2 by 2 complex-valued matrix function.
We also study the nonhomogenuous system. Miranda [13) has
studied the weakly coupled real elliptic system
n auLU+L~ak+Cu=f
k=l
giving a rather complicated condition which implies the bound for
solutions u,
4
When all the matrices ~ = 0, k=1, ••• ,n, the condition required by
Miranda reduces to eTce ~ -colel2 for anye E Rm. We will extend
this result to C being a complex matrix function such that
*C + C ~ -2coI (Corollary 6.3). Moreover, we prove the following
(Theorem 6.1):
*If there exist B > 0 and E > 0 such that C (x)B + BC(x) ~ -E in
D, then for all C2(D) nC(D) solutions u of Lu + Cu = f ,
Here K] =
eigenvalue
(:~ f/2-1
of EB .
and 2 (/3 )1/2K2 =~~ , where ~1 is the smallest
1 1
Results for systems are later used to yield maximum principles
and bounds for some higher order elliptic homogeneous and
nonhomogeneous equations. Our maximum principles include those of
Chow and Dunninger [2], [6] for real metaharmonic functions as a
special case. Various maximum principles for higher order elliptic
equations were also studied in the papers of Agmon [1], Duffin [4],
[5], Fichera [7], Payne [14], Scheafer and Walter [16], [17] and the
books of Miranda [12] and Sperb [19].
2. NOTATION AND A LIAPUNOV STABILITY THEOREM
Unless otherwise stated, all matrices considered in this paper
will be over the complex field. Let X be any mxn matrix. Its
transpose, complex conjugate and adjoint will be denoted by XT, Xand
* *-;;Tx (X =x ), respectively. For the sake of brevity, both Hermitian
positive definite and real symmetric positive definite matrices will
be named positive. Similar abbreviations hold for semipositive,
negative and seminegative definite matrices. The notations B > 0,
B ~ 0, B < 0 and B ~ 0 mean that the square matrix B is positive,
semipositive, negative and seminegative, respectively. The norm
11-110 D means the sup norm over D; thus for complex-valued vector,
I 12 I 12 1/2lIullo D ;= sup [uf x) I = sup ( u1(x ) + --- + um(x) .
'xeD xeD
The following well~known result in Liapunov stability theory will
be applied several times in this paper. We state this result as a
Lemma.
LIAPUNOV LEMMA. Let C be an mxm complex or real matrix.
(a) Assume that no eigenvalue of C has positive real part, and
moreover that the elementary divisors of C corresponding to
eigenvalues with vanishing real part are linear. Then there exist
*matrices B > 0 and E ~ 0 such that C B + BC =-E;
(b) If each eigenvalue of C has negative real part, then for any
*E > 0, there exists a unique B > 0 such that C B + BC = -E.
The proof of this Lemma, both for real and complex versions, can
be found in many papers, such as [11], [18] and [24].
5
3. MAXIMUM PRINCIPLES
Consider a second-order operator,
6
Here the summation convention is
(3.1)
in a bounded
a2
L=a .. (x)axalJ . X.
1 J
domain D in IRn .
a+ a.(x).:l.. ,1 VA.
1
a .. = a .. ,lJ J 1
employed. We assume that L is elliptic in D; i.e., for all XED and
(3.2)
holds.
a .. (x)y.y. > 0lJ 1 J
We also suppose that the coefficients a .. and a. are boundedlJ 1
and real-valued functions in D.
Now consider the following weakly coupled second-order elliptic
system,
Lu (x) + c 1 (xru, (x) = 0sSt K
or, in more brief matrix form,
s=1,2, ... ,m, in D ,
(3.3) Lu(x) + C(x)u(x) = 0 in D .
Here C(x) = (csk(x)) is an mxm complex matrix function and u is a C2
mx1 complex vector function. Associated with (3.3) is the following
characteristic equation of C,
(3.4) h(A) - IAI-ci = 0 .
THEOREM 3.1. Assume that there exists a constant matrix B > 0 such
that
(3.5) *C (x)B + BC(x) ~ 0 , XED .
Then for all C2 (D) nC(D) solutions u of (3.3), there exists a
positive constant K such that
(3.6) Ilullo D s K Ijullo an, ,
--------------------- ._----
7
f3 1/2Here K = (pm) ,where PI and Pm are the smallest and biggest
1
eigenvalues of B, respectively.
Proof: Define
where "."
*v = u Bu =ueBu = Bu.u =bksukus '
is the dot product in ~n defined by x.y = y*x
Then v is a nonnegative function and,
etc. ;
2_ a uk= ax.ax.
1 J~,ij
v . =~. = bk uk·u + bk uku .,1 VAl S ,1 S S S,1
_ a2vv .. = ax ax = bk Uk ., u + bk uku .. + 2Re{bk uk .u . }
,IJ ., S, IJ S S S, IJ S ,1 S, J1 J
aukuk . =ax.,1 1
where
and,
Lv = a .. v .. + a. v .IJ ,IJ 1 ,1
= bk a. 'Uk ..u + bk uka ..u .. + 2bk a ..uk.u .SIJ ,IJS S IJS,IJ SIJ ,IS,J
+ bk a,uk .n + bk uka.u .S 1 ,1 S S 1 S,1
= bk (Luk)u + bk uk(Lu ) + 2a ..u .•Bu .S S S S IJ ,1 ,J
* * 1/2 1/2= (Lu) Bu + u B(Lu) + 2a ..B u .•B u.IJ ,1 ,J
Thus,
(3.7)
since
vn
* * 1/2 1/2Lv = -u (C B+BC)u + 2a ..B u .•B u. ~ 0 ,1 J ,1 ,J
*C B + BC ~ 0 and a .. v .•v. ~ 0 for any vectors vI' v 2, .. , ,1 J 1 J
Therefore, by the maximum principle for the elliptic operator L,
we have
(3.8) vex) ~ max v(y)yean
for all xeD.
Suppose that {Pi}~=1 are the eigenvalues of B with PI ~ P2~•••~ Pm .
Since B > 0, we know that PI > ° and
P1Iu(x)12 ~ v(x) = u(x)*Bu(x) ~ Pmlu(x)12 .
Hence, from (3.8),
2 Pm 2[utx) I ~ -P- max [uf y) I ,
I ydJD
8
where K -_ ( ~ml J1/2and consequently, lIullo,D ~ K lIullo,aD ' fJ )
The Liapunov Lemma yields the following corollary.
I
COROLLARY 3.2. Let C(X) = r(x)I + C in (3.3), where r(x) ~ 0 in D
-and C is a constant matrix over the complex field. Assume that none
-of the eigenvalues of C has a positive real part, and moreover that
-the elementary divisors of C corresponding to eigenvalues with
vanishing real part are linear. Then there exists a positive constant
K such that for all C2(D) nC(n) solutions u of (3.3),
(3.6) Ilullo D ~ K Ilullo aD ., ,
Proof: By the Liapunov Lemma in Section 2, there exist matrices B > 0
-* -and E ~ ° such that C B + BC = -E ~ o. Since r ~ 0 in D, we get
* -* -C (x)B + BC(x) = 2r(x)B + C B + BC ~ ° .Now the result of this corollary follows from Theorem 3.1. I
Remark. By the Liapunov Lemma, we know that at least one positive
-* - -definite B, satisfying C B + BC = -E ~ 0, exists if the matrix C meets
-the assumption of Corollary 3.2. In fact, if C satisfies the condition
of Corollary 3.2 and has at least one eigenvalue with vanishing real
part, then there is an infinite number of positive definite B such
-* -that C B + BC = -E ~ 0 (see [11] and [18]).
Theorem 3.1 and Corollary 3.2 are generalized maximum principles
since the value of K in (3.6) may be larger than 1. The best
conceivable value of K in (3.6) , for any matrix C, is K = 1 , which
corresponds to the classical maximum principle.
THEOREM 3.3. (The Classical Maximum Principle)
(a) A sufficient condition that
9
(3.6)1
holds, for all
(3.9)
lIullo D s lIullo an, ,C2 (D) nC(D) solutions u of (3.3), is
*C (x) + C(x) ~ 0
(b) Assume that the variable matrix C = C(X) in (3.3) is normal
* *(i.e., C (x)C(x) = C(x)C (x), x E D), and all its eigenvalues have
nonpositive real parts for all XED. Then (3.6)1 holds for all
C2(D) nC(D) solutions u of (3.3).
Proof: (a) By choosing B = I in Theorem 3.1, (3.6) with K = 1
(i.e., (3.6)1) follows from the condition (3.9).
(b) Suppose
A1
(x) , A2
(x ) , ••• , Am(X)
are all the eigenvalues of C(X). Since C(X) is normal, there exists a
unitary matrix U(x) such that
and then (3.6)1 follows from (a).
2ReoX (x)m
*u (x)C(x)U(x) =
Therefore, by the assumption,
* *U (x)(C (x) + C(x»U(x) =
* *Hence C + C =UAU ~ 0
••
2Re\ (x)••••••
•oX (x)m
-. A(x) ~ o.
I
10
Remarks. 1. The condition (3.9) is also a "necessary" condition for
the proof of the classical maximum principle by the method imposed
here. In fact, if, in Theorem 3.1, (3.6) holds with K = 1, then P1
=
Pm' and so there exists a B > 0, with an m-multiple eigenvalue P > 0,
* *such that C B + BC ~ 0; hence B = PI, and then C + C ~ o.
2. Theorem 3.3 contains the result of Winter and Wong [23] for
real negative semidefinite C =C(x, u, Vu) as a special case; one
may view, for given u, C(x, u(x), Vu(x» as a matrix function C1(x).
EXAMPLE 3.4. For n = 2 , consider
Lu + [: ~] u = 0 , a, b, c, d E IR .
The associated characteristic equation,
oX2- (a+d)oX + (ad-bc) = 0 ,
has roots
Hence, by Corollary 3.2, the inequality (3.6) is valid provided one of
_ .. _.._--------- -_._---- ----
The inequality (3.6) is not valid for the general case, when
a + d > 0 or a + d = 0, ad - be ::;:; O. In fact, u = sinx Siny[_~J
solves the systems
and
~u + [_~ - ~] u = 0
in D = (O,~)x(O,~) and vanishes on an, but (3.6) does not hold.
Theorem 3.1 gives a sufficient condition for which (3.6) holds.
It raises some open questions as to whether Theorem 3.1 can be
extended to a more general system (3.3) with weaker restrictions on
11
the matrix C, and as to whether necessary conditions can be determined
so that (3.6) holds.
(3.7)
in the
Following from the inequality
* * * 1/2 1/2L(u Bu) ~ -u (C B + BC)u + 2a .. B u .•B u. ~ 0 ,1 J ,1 ,J
proof of Theorem 3.1, and from Protter and Weinberger's book
[15], are the following two maximum principles for system (3.3).
COROLLARY 3.5. If u E C2 (D) nC(D) is a solution of (3.3), and if
*u Bu attains a maximum in D for some positive definite matrix B such
*that C (x)B + BC(x) is negative semidefinite in D, then u is a complex
*constant vector in D. Moreover, if C (x)B + BC(x) is negative
12
which implies that
1/2and then, Bu. =,1
definite at some xED, or, if C(x) is invertible for some XED, then
u == 0 in D.
Proof: Under the assumption of this corollary, by the proof of
Theorem 3.1, inequality (3.7) holds. Thus, by the maximum principle
*for the second-order elliptic equation (see [15]), u Bu == constant.
Hence, from (3.7) again, we have
* * * 1/2 1/2o = L(u Bu) = -u (C B + BC)u + 2a ..B u .•B u. ~ 0 in D;1 J ,1 , J
* * 1/2 1/2u (C B + BC)u = 0 and a ..B u .•B u. = 0IJ ,1 , J
o and u . = 0 in D, for 1 ~ i ~ n Thus u is a,1
*complex constant vector in D. Moreover, if C (x)B + BC(x) < 0 at
* *some XED, then, from u (C (x)B + BC(x))u = 0 , we have u == 0 in D;
and if C(x) is invertible for some xED, then, from the system (3.3),
we still have u == 0 in D. I
Note that Corollary 3.5 actually holds even if D is unbounded.
COROLLARY 3.6.2 nC(l)) be a solution of (3.3). SupposeLet u E C CD)
* *that u Bu ~ M in D and that u Bu = M at a piont P E an for some
*positive definite B such that C B + BC is negative semidefinite.
Here M is a nonnega~ive constant. Assume that P lies on the boundary
of a ball in D, and that the outward directional derivative ~ exists
at P. Then
unless u is a complex constant vector such that
*a(u Bu)av = 2Re[u*B%;] = 2Re[(B1/ 2u)* a(~:/2u)J > 0
*u Bu == M
at P
equivalently,
o1B1/ 2ul
OV > 0 at P
unless u is constant and IB1/2ul =M1
/2
4. UNIQUENESS THEOREMS FOR SOME BOUNDARY VALUE PROBLEMS
As applications of the maximum principles in section 3, we can
prove uniqueness theorems for various boundary value problems.
As an example, consider the first boundary value problem for the
elliptic system (3.3). By Theorem 3.1, the problem
13
Lu(x) + C(x)u(x) = f(x)
ulan = g(x)
xED
where C satisfies the assumption of Theorem 3.1, has at most one
solution.
As a second example, we have uniqueness for the following mixed
boundary problem:
Lu(x) + C(x)u(x) = f(x)
. ufx) = g_ (x)
t ~(x) + :(x)u = g2(x)
in D C IRn
on r 1 '
on r 2 '
where v = vex) is a given outward direction on r2 ' and r2 = OD\r1
COROLLARY 4.1. Suppose u1 and u2 satisfy (3.3)N and (4.2) in a
bounded domain D C IRn and C satisfies the assumption of Theorem 3.1.
Assume that each point of r2
lIes on the boundary of a ball in D. If
1 2L is elliptic as defined by (3.1) and a(x) ~ 0 on r2
' then u =u ,
except when a = 0, r1
is vacuous and C(x) is singular for all xED,
. hi h 1 2. I tIn w IC case u - u IS a comp ex cons ant vector.
Proof: Define 1 2 Then u satisfiesu =u - u
(3.3) Lu + au = 0 in D
{:= 0on r
1,
(4.3)0 on r
2- + au =iJv
Choose positive definite B, as in the proof of Theorem 3.1, such that
* *C B + BC is negative semidefinite, and let v = u Bu ; then (3.7)
holds. If v is not a constant, by Corollary 3.5, a nonzero maximum of
14
v must occur at a point P on r2
;
Re[u*B~]
Hence,
and by Corollary 3.6,
> 0 at P.
Re [u*B(~ + au)] > 0 at PEr2 '
which contradicts the boundary condition (4.3). Thus *v = u Bu must
be a constant. From (3.7), by the proof of Corollary 3.5, u must be a
complex constant vector. Then, from (3.3) and (4.3), we know that
O. 1 2u= ,I.e., u =u in D, except when a = 0, r
1is vacuous and C(x)
is singular for all xED. I
Besides having applications to some boundary value problems with
bounded domain D, the maximum principle we obtained in section 3 can
also be used to establish uniqueness theorems for various boundary
value problems with unbounded domain D eRn In this case, the point
00 is called the exceptional boundary point, and an appropriate growth
restriction on the solution at 00 is required. Moreover, by using a
recent result of Hile and Yeh [10], we even obtain uniqueness for the
boundary value problem with an exceptional boundary set r such that
the Hausdorff dimension of r is less than n - 1.
As a third example of this section, we have uniqueness for the
following boundary value problem with unbounded domain D eRn
15
Lu(x) + C(x)u(x) = f(x) in D ,
{
u(x) = g(x) on an(4.4)
lu(x)1 is bounded in D .
COROLLARY 4.2. Let D be an unbounded domain contained inside a cone,
let L be given in D by (3.1), with
a.(x) = o(r-1) as x ~ 00 in D, i = 1, ... , n, (r = Ixl),
1
and with the uniform ellipticity condition
nxeD,yeR,
holding for some positive constants 8 and A. Suppose u1 and u2
satisfy (3.3)N ' (4.4) in D and matrix C satisfies the assumption of
Th 3 1 Then U1 =u 2 . Deorem . . In .
Proof: 1 2 *Let u = u - u and v = u Bu, where B, as in Theorem 3.1, is
*positive definite such that C (x)B + BC(x) is negative semidefinite
for all xeD. Then v is nonnegative, and the proof of Theorem 3.1
gives
* * 1/2 1/2-u (C B + BC)u + 2a ..B u.B u ~ 0IJ
= 0 on an,is bounded in D.
in D,
Hence, by the Phragmen-Lindelof Principle ([9], Corollary 2),
lim vex) = 0 , as x ~ 00 in D.
16
Therefore, the maximum principle (Theorem 3.1) implies that v =0 in
D · u1 =u
2; 1. e., in D. •
Remark. When L =~ (Laplace operator) in (3.3)N ' the growth
restriction of lui being bounded in Corollary 4.2 can be replaced by
the following weak restriction:
max [uf x) 12
1 · . f Jxl=r 0nn In =r~ log r
or
1~I~r lu(x)12
lim inf . . = 0n-2r~ r
(see [15]).
if n = 2 ,
if n;:: 3
As a last example, we prove uniqueness for the solution of the
following boundary problem without knowing the data on an exceptional
boundary set reaD (the domain D can be unbounded):
(3.3)N (~ + a i (x)~Ju(x) + C(x)u(x) = f'(x) in D,1
{
u(x) = g(x) on aD \ r ,(4.5)
\u(x)1 is bounded in D .
COROLLARY 4. 3 .1 2 2Suppose u and u are two C solutions of the problem
(3.3)N, (4.5); and assume that \ai(x)\, 1 ~ i ~ n, are bounded in D
and the matrix C(x) satisfies the assumption of Theorem 3.1. Let r be
a subset of aD such that, for each y on r, D is contained on one side
of an (n-1)-dimensional hyperplane passing through y. Assume also that
the Hausdorff dimension of r is less than n - 1. Then u1 =u
2in D.
Proof:
17
Let u = u 1- u 2 and v = u*Bu where B, as in Theorem 3.1, is
*positive definite such that C (x)B + BC(x) is negative semidefinite
over D. Then v is nonnegative, and the proof of Theorem 3.1 implies
that
{
(~ +
v(x)
v(x) ~ M in D, for some M> 0 .
Hence, by a result of Hile and Yeh ([10], Corollary 2),
in D,
Therefore,
v == 0
v == 0
on r.
on an.Thus, by the maximum principle (if D is bounded) or Phragmen-Lindelof
principle (if D is unbounded) for the second order elliptic equation,
we have v == 0 in D. This gives u1== u2 in D.
5. ESTIMATE OF K
I
P 1/2In Theorem 3.1, the constant K = ( pm ) in (3.6) depends on C;
1
in fact, (3 and (3 are the smallest and largest eigenvalues of B,1 m
where B > 0 is chosen so that
C (x)B + BC(x) ~ o.
The value of K in (3.6) is important in applications, such as in
numerical computation and estimation. The best conceivable value of
K, for any C, is K = 1, which corresponds to the classical maximum
principle; and Theorem 3.3 gives a sufficient condition (3.9) to
guarantee the classical maximum principle (K = 1) for solutions of
(3.3). However, for a general matrix C, the best possible value of K
can be larger than 1.
EXAMPLE 5.1. For the system
Luf x) + [:~~~ ~~~nu(x) = 0 ,
where a, b, c and d are complex-valued functions, by Theorem 3.3 the
classical maximum principle (3.6)1 holds if
18
[a b] * + [a b] = [2Re{~}cdc d (b+c)
which is equivalent to
b+c ] s 0 ,2Re{d}
(5.1) - 2Re{a} :;:; 0, Re{d} :;:; 0; and Ib + cl :;:; 4Re{a}Re{d}.
The following two examples can be used to compute the best choice
(f3 )1/2 [ b]of K = \ f3: when C = : d is a real constant matrix with
eigenvalues A :;:; A :;:; 0 (A , A not both zero).- + +-
EXAMPLE 5.2. Consider
(5.2) c E IR .
In order to determine the best K, we choose a positive definite
B := ~ ~J which minimizes
(5.3) K = ( o:+"y) +
(0:+')') -
A/ (a-y)2
I 2Iv· (a-,)
and satisfies
Without lost of generality, we can assume that
19
(5.4) a+1=1.
Then the problem can be reduced to the following equivalent problem:
Find (a, P) E R x ~ which
minimizes (2a-1)2 + 41PI 2
subject to: a > a2 + IpI 2
ECP+{i) s 2 ,
a2( E
2+4) + 41pI 2- 4a :;;; 0 .
Obviously, P = 0 is the best. Thus the problem reduces to
minimize (2a - 1)2
subject to: a > a2
a 2( E
2+4) - 4a :;;; 0 .
It is easy to conclude that
{' if lEI :;;; 2 ,('2,0)
(a,pl = [4 ~E 2 ,0) , if lEI > 2
K -- (~m)1/2Hence by (5.3) and (5.4), the best value of ~ for the1
system (5.2) is
{
1 ,K = ill
2 '
if lEI:;;; 2 ,
if lEI >2
EXAMPLE 5.3. Now we consider the generalized form of (5.2),
( 5 . 5) Lu + [- ~ _~] u = 0 , o ;:;: 0, E E IR
By going through the same process, we can reduce the problem of
minimizing K in (3.6) to finding B = ~ 1~a] > 0 , which
minimizes (201.-1)2 + 41PI 2
subject to: a > 0 ,
a - 01.2
- IP12
> 0 ,
(u-1) (p+p)aE + 4u(a-a2) - a2E - (1+u)2IPI 2 ~ U .
20
Using lagrange's method of multipiers and after going through a rather
complicated computation, we get
1(2,0)
2, E ::;40-,
(a,p)=
Hence we
is
«1+U)2 E2 + 2vCU (1+U)V/E2[E2+(U-l)2] (E2-4U)(U-1)E) E2>4u.
2{ E2(E2+(1_u)2) + 2VU (1+U)\tIE2[E2+(1-U)2] }
find that the best choice of K in (3.6) for the system (5.5)
1 2, E :;;;40-,
K =
[
E2[E2+(1-u) 2]
E2[E2+(1_u)2]
For a general real matrix C = [: ~] with eigenvalues A_ ~ A+ ~ 0,
A < 0, by matrix theory (see [20], Chapter 5), there exists an
orthogonal matrix P := [~ ~] and a real number EO such that
[A_ EO] = pTcp = [p~a+prc+prb+r2d Pia+rqc+sPb+S~d] .o A+ pqa+psc+rqb+rsd q a+qsc+qsb+s d
From 0 = pqa+psc+rqb+rsd, we have
Let
EO = pqa+rqc+spb+srd = (ps-rq)(b-c) = detP.(b-c) = ±(b-c).
TB = PBl. Then
CTB + Be = P { [> ::]\ + B1 [> ::]} pT
s 0
is equivalent to
21
or
EO _ b-cBy letting E = - -A--- = + ---A--- and
5.3 we have the following:
~ ° ,
~ f~ ]s 0 .
A+
U =-A---' from the Example
f3 1/2Conclusion: The best value of K = ( f3~ ) for C = [: ~] E IR2X2
in (3.3) with eigenvalues
(a+d)± A/ (a-d)2+4bcA± = 2 ~ 0, and (a+d) < 0,
is
1
if (b_c)2 ~ 4detC ;
Remarks. 1. By using
AA = deW ,- +
A + A = trC ,- +E~ = (b_c)2 = a2 + b2 + c2 +d2 _ (a+d)2 + 2(ad-bc)
= (trCTC)2 - (trC)2 + 2detC ,
(A - A )2 = (trC)2 - 4detC ,- +
we can express K in terms of the matrix C.
2. The condition for the classical maxim~m principle (K = 1) is
(b_c)2 :;; 4detC ,
and the equivalent condition for it is
(b+c)2 :;; 4ad (which is the same as (5.1)),
or
It is clear that not only the normal matrices, but also some other
matrices satisfy one of the above conditions.
22
In case C -- [ac
bd] E 1R2X2 h t 1 • t' 1as wo comp ex conJuga e elgenva ues
with nonpositive real parts,
(5.6) '± = (a+d) ± v/-2(a-d)2-4bC i1\ - : = J.L ± vi
( J.L,vEIR, )lp,:;;0, V>o
the problem of finding the best choice of K in (3.6) is very
complicated. The method we used above for real eigenvalues does not
succeed. Here we can only give a formula for an upper bound for the
best possible K.
From (5.6), we have
2J.L =a+d :;; 0,
2v = A/-(a-d)2-4bc > o.
Suppose that e .- a+ip E ~, (when 1>=[::]' p=~:] E JR') is an
eigenvector of C corresponding to the eigenvalue A =J.L+vi.+
Then
23
C(a-ip) = (~-vi)(a-ip),
which gives
or,
Ca = ua -vp,
Cp = J.Lp + va,
in matrix form C(a P) = (a P) [ J.L VJ .-v J.L
The general solution for (a P) in the above equation is
(5.7)
i. e.,
(5.8)
a-:j! b= .fJ +.:::.au ": V2'
_ .£. d-y- ~l + II 132 '
for any PI' P2 E ~\{o}.
Let
then
Since II > 0 and bc ~ 0, we see that a and p are independent over
~2. Hence the matrix (a P) is nonsingular and
C = (a P)[ J.L VJ(a p)-l-II J.L
B = (a p)-lTBl(a p)-l > 0;
CTB+BC = (a p)-lT[ J.L II]T(a p)-lTB (a p)-l-II J.L 1
+ (a p)-lTBl(a p)-l(a p)r J.L VJ(a p)-lI.-V J.L
Hence,
= (IV fJ)-lT( [ ~ IIJTB B [ J.L PJ ) ( fJ)-l.... ,... -II J.L 1 + 1 -II J.L a ,... .
CTB + BC :;:; 0 iff
(5.9)
The inequality (5.9) holds for Bl = I, and in this case,
B = (a p)-lT(a p)-l = [(a p)(a p)T]-l > o.
The characteristic equation for (a p)(a p)T is
--- - - - --- ------
2 2
[11 [ ,x-(al+Pl) -(ala2+PlP2)]
det ,xI-(a p)(a P)-J = det -(a a +P P) ,x_(a2+p2)1 2 1 2 2 2
= ,x2 _ tr(a p)(a p)T,x + det(a p)(a p)-l
\2 (2 2 2 2 \ 2= A + al+a2+Pl+P2)A + (a lP2-a2Pl)
Let .xl , ,x2 (0 < ,xl ~ ,x2) be the two eigenvalues of (a p)(a p)T.
-1 \-1Then the eigenvalues of Bare 0 <,xl ~ A 2 ' and
2 222 /22 222 2(a l+a2+Pl+P2) + V. (a l+a2+Pl+P2) - 4(alP2-a2Pl)
2222 /22222 2(a l+a2+Pl+P2) - V (a l+a2+Pl+P2) - 4(alP2-a2Pl)
By (5.8), we have
2 2 2 2 1 I II 2 21a l + a2 + PI + P2 =~ b-c -cPl+(a-d)p P2+bP ,v 1 2
1 2 2(a
lP
2- a
2P
l) = v(-cPl+(a-d)PlP2)+bP2)
Hence, by choosing Pl,P2E ~\{O} such that
-cp~ + (a-d)PlP 2 + bP~ ~ 0 ,
we obtain
1 [b-e] 1 / 2 2+- Iv. (b-c) -4v 1/(b+c) 2«(a-d) 22 v 2=~+(5.10) K2 v=
1 [b-e] 1 / 2 2 \b-cl - / 2 2-- Iv. (b-c) -4v Iv. (b+c) +(a-d)2 v 2V
Since the choice of Bl = I in (5.9) may not be the best choice,
the formula (5.10) for K2 is not the best one. From formula (5.10),
we obtain K = 1 when a = d and b = -c .
6. BOUNDS FOR SOLUTIONS OF THE NONHOMOOENEOUS ELLIPTIC SYSTEM
In section 3 we obtained maximum principles for some homogeneous
elliptic systems. Now we extend these results to the nonhomogeneous
24
25
elliptic system
Lu(x) + C(x)u(x) = f(x) , xED.
Under the restriction of C being a real matrix function such that
eTce ~ -colel2 in D, for some Co > 0 and for anye E Rm, Miranda [13]
obtained a bound for solutions of the elliptic system (3.3)N:
Ilullo D ~ Ilullo an + c~lllfilo D ., , ,
In this section, we obtain a similar result for more general complex
matrix functions C in the elliptic system (3.3)N
THEOREM 6.1. Suppose, for C =C(x), there exist two complex constant
*matrices B > 0 , E > 0 such that C (x)B + BC(x) ~ -E. Then for
all C2(D) nC(D) solutions u of (3.3)N' there exist positive constants
K1 and K2 such that
(6.1)
2 l/2m
1J1 12ILl'"1
, where PI and Pm are the
smallest and biggest eigenvalues of B, respectively, and ILl is the
-1smallest eigenvalue of EB .
Proof: Define 1/2v=B u; i.e. ,-112
U = B v .
First we will prove that (for v),
(6.2) Ilvllo D s IIvllo an + K IIfllo D, , , K > 0 .
It is sufficient to show that at an internal relative maximum of Ivl
(or of Ivl 2 = IBl / 2ul 2
= Bueu) ,
(6.3) [vtx) I s Klf(x) I
we have
= 0 ,
We assume v(x) ~ 0 at such a maximum; otherwise the inequality is
trivial.
2At a relative maximum of Ivl =Bueu,
a I 2ax vi = 2Re(Bu.u k)k '
and the matrix of second derivatives is negative semidefinite; i.e.,
[~ ] [ ]ax.ax = 2Re(v i·v k + v.v ik) ~ 01 k '" nxnnxn
Hence,
26
Since
Re(Bu.u k) = Re(v.v k) = 0 ,, ,
[Re(v .•v k + v-v 'k) ] ~ 0 0
,1, ,1 nxn
(a ik) > 0, we have that
a·k(v .•v k + Re(v.v ok)) ~ 0 °1 ,1 , ,1
Into this inequality we substitute the system (3.3)N and
Re(Bu.u k) = 0 to get,o ~ a·kv oeV k + Re(Bu.aoku 'k)
1 ,1 , 1 ,1
=a·kv .•v k + Re[Bu.(-a.u . - Cu + f)]1 ,1 , 1 ,1
1 *=a·kv .•v k - -2(0 B+BC)ueu + Re(Bu.f) .1 ,1 ,
Therefore,
Since
that
1 *~ -Re(Bu.f) ~ a·kv .•v k - -2(0 B+BC)u.u
1 ,1 ,
1 1 -1/2 -1/2~ 2 Eu.u =2(B EB )VeV
and EB- 1 have the same eigenvalues, it follows
f.J.p1 / 2 1vl If I ~ 1 Ivl 2
,m 2
and then (6.3) holds with
K =ILl
Hence we have proved (6.2).
27
By using (6.2) and substituting 1/2v = B u, we have
so (6.1) holds with
_ (,BmJ
l / 2
K1 - l ,01and
2 ,01/2m I
From the Liapunov Lemma, the following corollary is easily
obtained:
-COROLLARY 6.2. Assume that C(x) = r(x)I + C in (3.3)N ' where r ~ 0
-in D and each eigenvalue of the complex constant matrix C has negative
2 -real part. Then, for all C (D) nC(D) solutions u of (3.3)N' there
exist positive constants K1 and K2
such that (6.1) holds.
COROLLARY 6.3. (a) A sufficient condition that
(6.4) lIullo D s lIullo an + c;ll1 f llo D, , ,2 -holds, for all C (D) nC(D) solutions u of (3.3)N' is
*(6.5) C (x) + C(x) ~ -2coI < 0, for some Co E R
(b) Suppose that the variable matrix C = C(X) is normal, and all
its eigenvalues {A.(X)}~ 1 have uniformly negative real parts in D;1 1=
i.e., ReAi(x) ~ -co < 0 in D, for 1 ~ i ~ m. Then (6.4) holds for all
2 -C (D) nC(D) solutions u of (3.3)N .
Proof: (a) By choosing B = I in Theorem 6.1, the inequality (6.4)
follows from the conditon (6.5).
(b) Since C(x) is normal, there exists a unitary matrix U(x)
such that
28
[
2Re\ (x)
U* (x) [c* (x) + C(x)]U(x) = •••••
·2ReA (x)m
*Hence C (x) + C(X) ~ -2coI; and then (6.4) follows from (a).
7. APPLICATIONS TO HIGHER ORDER EQUATIONS
I
Note that the results we obtain are valid for complex systems
which may be considered generalizations of some higher order complex
equations. For example, consider a 2m-order homogeneous equation of
the form
(7.1)
and the nonhomogeneous equation of the same form,
Pu = F.
Here ao ' aI' 0 •• ,am_ 1and F are complex-valued functions, ~ := L +
rex) where r ~ 0 in D and L is the elliptic operator defined by (3.1);
Let m-l= tu , ••• , urn = I u;
then the equations e7.1) and e7.1)N reduce to the equivalent systems
(3.3) and e3.3)N ' respectively, where
0 -1 u1
0o -1 u
2 •e7.2) C = r I • • and f •+ • • u = = •• • • 0
0 -1 ••ao a
1•••••a um_ 1
Fm-l
Associated with each system is the characteristic equation
(7.3) h(A) =(_A)m + a 1(-A)m-1 + 00. + a (-A) + a = O.m- 1 0
29
Hence, under the same assumptions on C in (7.2), Theorem 3.1 and
(7.4)
Theorem 3.3 hold, but (3.6) changes to
(m-1. )1/2 (m-1. )1/2
sup 1 !tJu(x)1 2~ K sup 1 ItJu(x)1 2 ,
xED j =0 xeaD j =0
also, under the same assumptions on C in (7.2), Theorem 6.1
(K ~ 1);
holds, but
-1 ]a
1(x)
(6.1) changes to
(m- 1 . 2)1/2 (m-1. 2)1/2
(7.4)N sup .1 ItJu(x) I s K1sup .1 \tJu(x) I + K211Fllo an'xED J=O xeaD J=O '
In the case of m = 2 and r =0 in D , the matrix in (7.2) is
a ± v/a2-4awith eigenvalues A± = 1 ~ 1 0
Hence the requirement for Theorem 3.1 holding is that there exists
B := [; ~J > 0 such that
= [ 2Re(sao)
qaO+sa 1
- p
-1 ]
a1(x)
which is equivalent to the existence of p,q e ~, s e [ such that
(7.5){
p > 0
pq _ Is 12 > 0and
{
Re(sao) ~ 0, R~(qa1;S) ~ 0
4Re(sao)oRe(qa1-s) ~ !qao+sa1-pI2.
If we choose s = 0 then
B = [~ ~J 'K2 _ max(p,q)
- min(p,q) ,
and (7.5) becomes
{
p > 0
q > 0and
30
So for the 4th-order equations,
(7.6)
and
we have
in D ,
in D ,
COROLLARY 7.1. Assume that Re{a1(x)} ~ 0 in D and ao > 0 , then
(a) for all complex-valued C4 (D) nC2(D) solutions u of (7.6),
I 12 I 12 1/2 1 -1 I 12 I 12 1/2sup( u + Iu) ~ V max(ao,ao ) sup( u + Iu) .D ~
(b) if rex) ~ -ro < 0 in D, then for all complex-valued
C4 (D) nC2(D) solutions u of (7.6)N'
s~p(luI2+IIUI2)1/2s v/max(ao,a~1)[sag(luI2+IIUI2)1/2 + r~11IFllo,~].
m-1When {aj}j=O are complex constants, by applying Corollary 3.2
and Corollary 6.2 we have the following corollary for the 2m-order
equations (7.1) and (7.1)N'
COROLLARY 7.2.
the conditions,
Suppose the roots {A.}.m1 of (7.3) satisfy both of1 1=
(i) Re A. s 0 ;1
(Li ) if Re A. = 0, then the corresponding elementary divisor1
is linear.
Then,
(a) there exists a positive constant K such that (7.4) holds for
all complex-valued regular solutions of (7.1).
31
(b) if ReA. < 0 ,1
1 :::; i :::; m ; or rex) :::; -r < 0o in D, there
exist positive constants K1 and K2 such that (7.4)N holds for all
complex-valued regular solutions of (7.1)N'
Remark. Corollary 7.2 contains the result of Chow and Dunninger [2],
[6] for real metaharmonic functions as a special case. The first part
of the Corollary 7.2 was obtained in [2] for the case of L =~
(Laplace operator) and ao ' a1
' ••• ,a being real constants.m-1
REFERENCES FOR PART I
[1] S. AGMON, Maximum theorems for solutions of higher order
elliptic equations, Bull. Amer. Math. Soc., Vol.66, (1960),
77-80.
[2] S. N. CHOW and D. R. DUNNINGER, A maximum principle for
n-metaharmonic functions, Proc. Amer. Math. Soc., 43 (1974),
79-83.
[3] M. A. DOW, Maximum principles for weakly coupled systems of
quasilinear parabolic inequalities, J. Austral. Math. Soc., 19
(1975), 103-120.
32
[4] R. J. DUFFIN, On aquestion of Hadamard concerning
super-biharmonic function, J. Math and Phys., 27 (1948), 253-258.
[5] R. J. DUFFIN, The maximum principle and biharmonic functions, J.
Math. Anal. Appl., 3 (1961), 399-405.
[6] D. R. DUNNINGER, Maximum principles for solutions of some fourth
order elliptic equations, J. Math. Anal. Appl., 37 (1972),
665-658.
[7] G. FICHERA, EI teorema del massimo modulo per l'equazione
dell'elastostatica tridimensionale, Arch. for Rat. Mech. and
Anal., 7 (1961), 373-387.
[8] G. N. HlLE and M. H. PROTTER, Maximum principles for a class of
first-order elliptical systems, J. Diff. Equ., 24 (1977),
136-151.
[9] G. N. HlLE and R. Z. YEH, Phragmen-Lindelof principles for
solutions of elliptic differential inequalities, J. Math. Anal.
Appl., 107 (1985),478-497.
33
[10] G. N. HILE and R. Z. YEH, Exceptional boundary set for solutions
of elliptic partial differential inequalities, Indiana Univ.
Math. J., 35 (1986), 611-621.
[11] J. L. MASSERA, Contributions to stability theory, Ann. of Math.,
(2) 64 (1956), 182-206. MR18 , 42.
[12] C. MIRANDA, "Partial Differential Equations of Elliptic Type,
Second Revised Edition," Springer-Verlag, Berlin, 1970.
[13] C. MIRANDA, SuI teorema del massimo modulo per una classe di
sistemi ellitici di equazioni de secondo ordine e per ie
equazioni a coefficienti complessi, 1st. Lombardo Accad. Sci.
Lett. Rend., A 104 (1970), 736-745.
[14] L. E. PAYNE, Some remarks on maximum principles, J. Analyse
Math., 30 (1976), 421-433.
[15] M. H. PROTTER and H. F. WEINBERGER, "Maximum Principles in
Differential Equations," Springer-Verlag, Berlin, 1984.
[16] P. W. SCHEAFER, On a maximum principle for a class of
fourth-order semilinear elliptic equations, Proc. Royal Soc.
Edinburhg, 77A (1977), 319-323.
[17] P. W. SCHEAFER and W.WALTER, On pointwise estimates for
Metaharmonic functions, J. Math. Anal. Appl., 69 (1979),171-179.
[18] J. SNYDERS and M. ZAKAI, On nonnegative solutions of the
equation AD+DA'=-C, SIAM J. Appl. Math., 18 (1970), 704-714.
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.._.--_. - --- -----
Press, New York, 1981.
34
[20] G. STRANG, "Linear Algebra and Its Applications," Second Edition,
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PART II
SCHAUDER ESTIMATES AND EXISTENCE THEORY FOR
ENTIRE SOLUTIONS OF LINEAR FOURTH ORDER ELLIPTIC EQUATIONS
35
1. INTRODUCTION
We investigate here solutions in ~n of the linear, fourth order,
elliptic, variable coefficient, nonhomogeneous partial differential
equation
36
(1.1)
n.- L a. °klrP.. k 1-1 IJ x.xoxkx II,J,,- IJ
n+ L b.okrP.. k IJ x.x.xkI,J, =1 1 J
n n+ L c.. ¢ + L d .¢ + e¢ = f,
o • 1 IJ X.X. . 1 1 X.1, J= 1 J 1= 1
and the related homogeneous equation,
(1.2) up = O.
Following the prevailing terminology in the literature, we refer to
solutions of these equations defined in all of ~n as entire solutions.
We shall develop Schauder-type a priori estimates for entire solutions
with prescribed behavior required at infinity. These estimates, of
perhaps independent interest themselves, lead to an existence and
uniqueness theory for entire solutions with certain behaviour required
at infinity whenever the operator L can be separated as
(1. 3)
Here
n fi n aL .- a .D2
+ b .D + c .- Lao ° a a + Lb. ~ + c (s=1,2)s s s so. 1 s 1 J x 0 x 0 • 1 s1 VlI.. S1,J= 1 J 1= 1
are two second order elliptic operators.
The coefficients a , b , c of L (s = 1, 2) are assumed to bes s s s
Holder-continuous in ~nU{oo}, and, as x ~ 00, the matrices a approachs
the n x n identity matrix I, the vectors b approach the 1 x n zeros
37
vector, and the scalars cs approach zero. Thus Ls (s = 1, 2) approach
the Laplace operator ~ and L = L2L1
approaches the biharmonic operator
~2 near infinity. We also assume the operators L are uniformlys
elliptic in ~n, and c ~ 0 if n ~ 3, c =0 if n = 2 (s = 1, 2). Withs s
these assumptions, and for n ~ 5, we shall establish a one-to-one
correspondence between the entire solutions of (1.1) that are O(lxl m)
near infinity and the polynomial solutions to the biharmonic equation
of degree no greater than m; when f =0, this result is an extension
of a Liouville-type theorem.
For dimension n ~ 4 we have similar results, but, like the case
of n = 2 in [2, 9] for second order equations, there are complications
which make the situations in four, three and two dimensions more and
more difficult. As Friedman, Begehr and Hile showed in [9, 2], when
n = 2 is the order of the equation, there need not exist an entire
bounded solution of Ll~ = f, even when L1
is the Laplace operator and
f has compact support in ~2. Begehr and Hile also proved that there
exists a unique entire solution of L1¢ = f with some growth
restriction at infinity and the condition ~(x) - 1logixl ~ 0 as
x ~ 00, for some constant 1. We obtain a similar result for the
~ .nsr(x) ~ 0 as x ~ 00, for some constant numbers
that ~(x) -
fourth order equation (1.1) when n ~ 4; our condition at infinity is
4-n1
s=o
~o' constant vector 71
and constant symmtric matrix ~2' Here r is a
fundamental solution of the biharmonic equation.
Several authors, for example [2, 5, 9, 10, 13, 15], have
investigated existence and/or uniqueness questions for entire
38
solutions of linear second order elliptic equations and systems.
There has also been considerable work on entire solutions of nonlinear
and quasilinear second order elliptic equations and systems, as for
example in [3, 12, 14, 16, 18-20]. The usual result obtained for a
linear or nonlinear second order equation is the traditional statement
of Liouville's Theorem, namely, a bounded entire solution is
necessarily constant. For the higher order case, Douglis and
Nirenberg proved a Liouville-type theorem for certain homogeneous
systems with constant coefficients in [4]. Liouville Theorems
for polyharmonic and some metaharmonic functions can be found in
[1, 6, 8],
Results analogous to the ones presented here for fourth order
equations appear to be most closely related to those of Begehr and
Rile [2] for second order equations. The Schauder estimates for
entire solutions are obtained from analogous Schauder estimates for
bounded domains as found in [4].
2. NORMS AND A SCHAUDER ESTIMATE FOR ENTIRE SOLUTIONS
For vectors v, w in ~n we let VeW denote the dot product of v and
w, and Ivl the Euclidean norm of v.
= (b .. ) in ~nxn we defineIJ
For n x n matrices A = (a .. ), BIJ
i ,j1 a .. b .. ,
IJ IJIAI = (AeA)I/2 = { 1 a~.}1/2 .
. . IJ1, J
) . nxnxnSimilarly, for tensors A = (a. 'k)' B = (b" k In ~ and C =IJ IJ(c
i j kl),D = (d
i j kl)in ~nxnxnxn we define
------ ---- -----------
and
A.B = 1 a. 'kb"k. . k 1J 1J1, J ,
I I ~ 2} 1/2A = { LJ a i jki ,j,k
39
C.D =
Let
~ I I 2 1/2LJ Cijkldijkl' e = { 1 ci j kl}i,j,k,l i,j,k,l
(The inequalities IA.al ~ IAIIal and IC.DI ~ lei IDI hold.) We let x
= (xl' x2' ... , xn)T denote a point in ~n, n ~ 2, and n a domain in
~n. For ~ = ~(x), a real valued function defined in n, we let D¢,
D2~, D3~ and D4~ represent the n X 1 vector of first derivatives of ~,
the n X n matrix of second derivatives of ~, the n X n X n tensor of
third derivatives of ~, and the n x n X n X n tensor of fourth
derivatives of ~,respectively. Thus,
D¢ .- (~x ' ~x ' ... ~x )T, D2~:= (~x.x.)nxn '1 2 n 1 J
D3~ ._ (~) D4~ := (~ ) .x.x'Xk nxnxn x.x,xkx I nxnxnxn
1 J 1 Ja=(a.· k l) ,b=(b..k) ,c=(c .. ) ,d=(d.) ,eIJ nxnxnxn IJ nxnxn 1J nxn 1 lxn
be functions defined in n, with values in ~nxnxnxn, ~nxnxn, ~nxn, ~n,
and ~,respectively. The tensors a, b and the matrix c are always
assumed to be symmetric; i.e.,
a i j kl = au(i)u(j)u(k)u(l) , for any permutation u of {i, j, k, I},
bi jk = b ikj = bj i k = bj ki = bki 1 = bkj i ' and a i j = a j i
We consider the fourth order linear partial differential operator L,
defined by
(2.1) ~ := a.D4~ + b.D3~ + c.D2~ + d.~ + e~
n n= 1 a. 'kl~ + 1 b. 'k~. , k 1-1 1J x .x ,xkx1 .. k 1 1 J x .x .XkI,J, , - 1 J I,J, = 1 J
n n+ 1 c ..~ + 1 d. ~ + e~ .
. . 1 IJ X.X. 1'--1 1 Xl'I,J= 1 J
Associated with the operator L are the homogeneous equation,
40
(2.2) up =0,
and the nonhomogeneous equation
(2.3) up = f ,
where the right-hand side f is assumed to be a real valued function
in n.
Using the notation of [11], for points x, y in 0, we define
dx := dist(x, an), dx,y:= min(dx' dy)'
and for u = u(x) a real valued, vector valued, matrix valued, or
tensor valued function in 0 we define the following norms (where ff E
JR, 0 < 0: ::; 1):
(2.4)
(2.5)
(2.6)
lul~~~ := sup dUlu(x)1 ,, XEO x
(o) [uCx) - u (y) I[u]O 0:'0'- sup (dx y)U+O: -----0:-
" x, YEO' Ix - y Ilui (e') := lui (U) + Iu l (o) .
0,0:;0 0;0 0,0:;0
By a classical solution of up = f, we mean a solution ~ with ~,
D¢, D2~, D3~, D4~ continuous. It is known (see [7], Chapter 9) that
if a, b, c, d, e and f are Holder-continuous with exponent 0: in n,
o < 0: < 1, and ~ is a solution of (2.3), then~, D¢, D2~, D3~ and D4~
are all Holder-continuous with the same exponent 0: in O. So, under
the condition that all cofficients and f are Holder-continuous, any
solution of (2.3) is a classical solution. Douglis and Nirenberg have
obtained a Schauder estimate for certain higher order elliptic systems
(Theorem 1, [4]). Applying their result to the single equation (2.3),
41
we get the following basic Schauder estimate:
LEMMA 2.1 ([4]). Let 0 be a bounded domain in ~n, n ~ 2, and let
~ E C4(0), with 1~1~?h finite, and with ~ a classical solution of the,nonhomogeneous equation (2.3) in O. Assume 0 < a < 1, that If\(4)
o,a;O
is also finite, and that for some nonnegative constant A,
I 1(0) I 1(1) I 1(2) I 1(3) I 1(4)
(2.7) a o,a;O' b o,a;O' c o,a;O' d o,a;O' e o,a;O ~ AAlso assume the symmetric tensor a satisfies the ellipticity
condition
n(2.8) E aijkl(x)eiejekel ~ Alel
4,for all x E 0, eE ~n,
i,j,k,I=1
for some positive constant A. Then for some universal nonnegative
constant M= M(n, a, A, A) (independent of ~) we have the bound
(2.9) I~(x) 10(?A + IDr/> I (~) + ID2~1 (~) + ID3~1 (~) + ID4~1 (4).,I' 0,0 0,0 0,0 0,0:,0
~ M(n, a, A, A){I~I~~n + Ifl~~~;o}.
(Throughout this chapter we denote generic constants by M( ), where
inside the parentheses are listed the entities that determine M.)
We will derive an analogue of Lemma 2.1 for the case n = ~n.
For this purpose it is useful to define the following norms, where
again U E ~, 0 < a ~ 1, and now u js a function defined on all of ~n.
Let
Ilullu := sup (l + Ix I)-ulu(x) I,x~n
[utx) - uf y) I(1 + [x] )-a-o: ----
Ix _ ylasup
nx,y~
o<lx-yl~(1+lxl)/2
Ilull := lIull + Ilull( )o ,« a U ,0:
Ilull(u,a) .-(2.11)
(2.12)
42
The norms (2.10)-(2.12) were first defined by H. Begehr and G. N. Hile
in [2], and may be viewed as analogous to the norms (2.4)-(2.6),
where the distance dx' vanishing as x approaches an, has been replaced
by the quantity (1 + Ixl)-l, which likewise vanishes as x approaches
the boundary point at infinity of Rn. The condition 0 < Ix - yl ~
(1 + Ixl)/2 of (2.11) is a technical convenience, ensuring that y
approaches infinity along with x, and more or less in the same
direction.
DEFINITION 2.2. We say that a real valued, vector valued, matrix
valued, or tensor valued function u defined on all of Rn is in the
space
(i) Bu if and only if lIullu is finite,
(ii) B if and only if lIull is finite.u,a u,a
We see that if u E Bu' then lu(x)1 = O(lxl u) as x ~ 00. The
space Bo consists of bounded functions in IRn, with lIullo reducing to
the ordinary supremum norm. If u E B then the Holder quotient,u,a
lu(x)-u(x)l/lx-yla, is O(lxl u-a) (with the restriction 0 < Ix-YI ~
(1+lxl)/2). In particular, functions in B are uniformly Holder-u,a
continuous with exponent a on compact subsets of Rn.
We gether some miscellaneous observations regarding the spaces Bu
and B into the following lemma. The proof of the lemma can beu,a
found in [2].
LEMMA 2.3. Let u, T E R, 0 < a, ~ ~ 1, and let u and v denote
functions defined on IRn (both scalar valued, both vector valued, both
43
matrix valued, or both tensor valued). Then
U s r , 0 < {J s a s 1 ~ B C B (J' Ilull (J s lIull ;a ,« T,,., T,,., o ,«
(c) u E Bu ' v E B". ~ UeV E BU+T ' lIuevllu+T ~ lIullu IIvll". ;
(d) u E B , v E B ~ UeV E B ,u,a T,a U+T,a
(a) U ~ ". ~ B... C B , lIull ~ Ilull ;v r r U
(b)
Moreover,
(e) if ~ is a real valued function in Cl(~n), with ~ E B ,U-l
then 1I~II(u,1) is finite, and in fact
1I~II(u,l) s M(u)II~llu_l
Finally,
~ K for all m, for some nonnegative constant K.
that we have a uniform bound, Ilu IIm u,a
Then there
(f) let {um}: =1 be a sequence of functions in Bu,a ' with all
n nxn nxnxnum's having values in the same space ~, ~ , ~ , ~ , or
nxnxnxn~ , and suppose
exists a subsequence {u }, and a function u in B , withmk u,a
also Ilull ~ K, such that u ~ u in the norm of any spaceu,a mk
B or B (J with". > u, 0 < {J < a.T T,,.,
We now present the following modification of Lemma 2.1, where nnis replaced by ~ and the norms (2.4)-(2.6) by newly-defined norms
(2.10)-(2.12).
THEOREM 2.4 (Schauder Estimate in ~n). Let ~ E C4(~n), n ~ 2, with
II~II finite for some U E ~, and with ~ a classical solution in ~n ofU
the nonhomogeneous equation (2.3). Assume 0 < a < 1, that IIfll isu-4,a
also finite, and that for some nonnegative constant A,
44
(2.13)
Also assume the tensor a is symmetric and satisfies the ellipticity
condition
n 41 a. ·kl(x)e.e.ekel ~ Aiel , for all x, e E ~n,
. . k 1-1 1 J 1 J1, J, , -
for some positive constant A. Then for some nonnegative constant M=
M(n, a, u, A, A) we have the bound
(2.14) 114>llu , l + 1ID4>lIu_1 , 1 + IID24>lIu_2 , 1 + IID3
4>lIu_3 , 1 + IID44> lIu_4 ,a
:s; M(n, a, o; A, A){1I4>11 + IIfll 4 }.(J U- ,a
Proof: First, we consider the case Ixl :s; 5. We apply Lemma 2.1 to
the domain
o := {z E ~n : Iz I < 9}.
The restrictions Ixl :s; 5 and 0 < Ix - yl :s; (1 + Ixl)/2 imply that
(2.15) 1 :s; dx' dy' dx,y :s; 9.
In order to check that conditions (2.7) hold in 0, we observe first
that
and hence
C2.18) I I (4)e 0 .n ~ MCa)A.,a,
45
In a similar manner we derive the estimates
C2.19) Idl~3~.n ~ MCa)l\dll_3 a ~ MCa)A,, , ,
Icl~~~;n s MCa)l\ cll_2,a s MCa)A,
Ibl~l~.n s MCa)l\bll_1 a ~ MCa)A,, , ,
lal~~~;n s MCa)llallo,a ~ M(a)A;
C2.20) It/>I~~n ~ MCu)IIt/>lIu' IfI6~~;n ~ M(a,u)llfllu_4,aTherefore Lemma 2.1 applies, and from C2.9), C2.20) we obtain, for
x, YEn, an estimate of the form
C2.21)
But for
1 ~ 1 +
C2.22)
It/>Cx) \ + d I~Cx)1 + d2ID2t/>Cx)I + d3ID3t/>Cx)I + d4\D4t/>CX)\x x x x
4 ID4t/>Cx) - D4t/>Cy) I
+ dx+~ a ~ MCn,a,u,>.,AHIIt/>lIu + II fl\u_4 a}', [x - yl '
Ixl ~ 5, 0 < Ix -y! ~ C1 + Ixl )/2, we have C2.15) and also
Ixl ~ 6, and therefore we may write C2.21) as
C1+lxl)-ult/>Cx)I + C1+lxl)1-ulDt/>Cx)I
+ C1+lxl)2-uID2~Cx)1 + C1+lxl)3-uID3~Cx)1
ID4~Cx) _ D4~Cy)1+ C1+lxl )4-u ID4tjJCx) I + C1+lxl )4-u+a ------
Ix _ yla
~ MCn,a,u,>.,A){II~1I + IIrll 4 i.U u- ,a
We next consider the case Ixl ~ 5. We define
C2.23) p := u + Ix I)/6, Ix I = 6p - 1,
and we now apply Lemma 2.1 to the domain
n := {z E ~n : p < Izi < lOp}.
We again check conditions C2.7) in n. For z, ~ E n we have
d ,d~, d ~ ~ 5p, P ~ l+lzl, 1+1~1 ~ 11p.z ~ z, ~
As in (2.16) and (2.17) we derive
1t;16~n:::; 54 sup O+lzl)4Ie(z)1 :::; 5411elL4 :::; 54A,
[e](4) (\ s (5p)4+a(lIe ll 4 (J' sup O+lzl )-4-ao,a;u (- , ) zen
+ 2-a!le1L4 sup (1+ Iz I)-4-a)
zen
s M(a)lIeIL4 a s M(a)A.,
A similar analysis shows that (2.19) continues to hold, but in place
of (2.20) we obtain
46
+
(2.24) ItPl6?n = sup 1¢(z)1 :::; (llp)(J' sup O+lzl)-(J'ltP(z)I :::; M((J')p(J'lltPll ,, ZEn zen (J'
IfI6~~;n :::; M(a)lI fll_4,a
~ Mca.UlpU{ sup Cl+lzll,-ulf(zl!
supZ,~En
0<lz-~I:::;(1+lzl)/2
:::; M(a,(J')p(J'lI fll(J'_4,a .
We now apply (2.9) to our particular point x, also using (2.24), to
obtain, for yEn,
(2.25) \tP(x) I + d 1D¢(x)I + d2 ID2tP(x)I + d3 ID3tP(x)Ix x x
4 4 4 a ID4¢(x)
- D4
tP(y ) I+ d ID tP(x) I + d + a
x x x,y [x _ yl
:::; M(n,a,(J',>.,A)p(J'{II¢1I + IIfll 4 }.(J' (J'- ,a
Now we have 1 + Ixl = 6p, and if 0 < Ix - yl :::; (1 + Ixl)/2, then
yEn and p :::; dx' dy ' dx,y :::; 5p. Thus (2.25) implies (2.22) also for
Ixl ~ 5, 0 < Ix - yl :::; (1 + Ixl)/2.
Upon taking supremums in (2.22), we conclude that
(2.26) 114>lIu + 1ID4>lIu_1 + IID24>llu_2 + IID34>lI
u_3 + IID44>lIu_4 a,
:;:; M(n,a,u,.x,AHII4>lIu + II f llu_4 aL,By Lemma 2.3(e) we have 114>11 ,.. 1 :;:; M(u)IID4>1I I' which implies we may
(v, ) u-
add the term 114>11 to the left-hand side of (2.26). SinceU,I
D24>E B 2' we may also apply Lemma 2.3(e) to each of the derivativesu-
ux. and conclude that 1ID4>II(U_I,I) s M(u)IID24>llu_20 Similarly, since1
D34>E B 3 and D44>
E B 4' we also obtain thatu- u-
IID24>II(U_2,1) :;:; M(u)IID34>lIu_3 and IID34>II(U_3,1):;:; M(u)IID
44>lIu_4'
Thus (2.26) implies our desired inequality (2.14). I
3. BOUNDS FOR ENTIRE SOLUTIONS OF THE NONHOMOGENEOUS BIHARMONIC
EQUATION
Before further discussing entire soluitons of the general fourth
order elliptic equation, it is useful both as motivation and as a
mathematical aid to study entire solutions of the nonhomogeneous
biharmonic equation,
47
(30U2/}. 4> = f.
In Rn, let r denote the fundamental solution of the biharmonic
equation,
(3.2) rex)
and for a real
1 IxI 4-
n , 1°f n"" 5 32(2-n) (4-n)w ~ or n = ,n
._ - _1_ l og[x] if n = 4,4w
4
~ Ixl 2(loglxl - 1), if n = 2;w2
valued function f on Rn define the Z-potential of f as
(3.3) Zf(x) := J r(x-y)f(y)dy.IRn
48
(3.4)
Here w denotes the surface area of the unit ball in IRD.n
We have the following estimates on the Z-potential.
LEMMA 3.1. let f be a real valued and measurable function on IRn
(n ~ 2), in the space B ,T > 4. Then-T
(a) for x E IRn the integral Zf(x) exists and
(1+IX I)4- 7" . f, 1 n~5, r-cn ,
(1+lx!)4-nl og(2+lxl), if n~5, T=n,
IZf(x) I :;;; M(n,T)lIfll O+lxl )4-n , if n=3 or ~5, T>n,-7"
log(2+lxl) , if n=4,
O+lxl )2 l og(2+lxl) ,if n=2;
(b) for Ixl ~ 1 we have,
(i) if n = 4, 4 < T < 5,
(3.5)"10
= IZf(x) + 4w4l0glxl I
:;;; M(7")llfll (1+lx\)4-T l og(2+lx\);-7"(ii) if n = 3, 4 < T < 5,
(3.6) IZf(x) - "Ior(x) + "Ilenr(x)I = \Zf(X) + :~ Ixl - ~I
:;;; M(T)lIfll O+lxl)4-7";-T
(iii) if n = 2, 4 < 7" < 5,
(3.7) IZf(x) - "Ior(x) + "Ilenr(x) - t "I2en2r(x)I
"10 2 "I IOX
= Zf(x) - -- Ixl (loglxl-1) + ---- (2loglxl-1)8~ 8~
lcglxl'Y2- -.811"
Ixl2
~ M(T)lIfli (1+lxl)4-T l Og(2+lxP;-T
where
loglxl
49
(3.8)
'Yo := J~n f(y)dy, "11 := J~n yf(y)dy,
'Y2 := Jn yyTf(Y)dY = J n (Y~ Yl~2Jf(Y)dYIR ~ lY1Y2 Y2
are a constant scalar, vector and matrix, respectively.
Proof: (a) By applying the estimate
to the integral (3.3), we obtain an inequality
(3.10)
where we define
(3.11)
(3.12)
(3.13)
:= J Ir(x-y)l(l+lyl)-Tdy,ly-xl~(1+lxl)/2
:= J [I'(x-y) I (1+lyl)-Tdy,Iy-x I,=2(1+ Ix I)
:= J Ir(x-y)I (1+lyp-Tdy.
(1+lxl)/2~ly-xl~2(1+lxl)
In I1(x),
we observe that
(3.14) Iy-xl ~ (1+lxl)/2 ~ (1+lxl)/2 ~ l+lyl ~ 3(1+lxl)/2,
which gives
(3.15)
{
(1+IX\) 4- r ,~ M(n,r) 4 r
(l+lxl) - log(2+lxl),
if n = 3 or n ~ 5,
if n = 2 or 4.
50
[y-x] ~ 2(1+lxl) =::;> [x] ::; Ix-YI/2 ::; IYI,
and therefore
if n = 2 or 4.
if n = 3 or n ~ 5,
(3.17) I2
(x) s M(r)J Ir(x-y) Ilx-yl-rdyly-xl~2(1+lxl)
~ M(r)J Ir(z)llzl-rdzIzl~l+lxl
{
( 1+ IX I)4-r ,
::; M(n,r) 4 r(l+lxl) - log(2+lxl),
In 13
( x ) ,
(3.18) (1+lxl)/2 ~ !y-xl ~ 2(1+lxl) =::;> Iyl ::; 2+3Ixl,
and moreover, if n = 3 or n ~ 5,
(3.19)
(3.21)
(3.20)
and if n = 4,
-Iog2 ~ log ~::; loglx-yl = 4w4Ir(x-y)I
~ log 2(1+lxl) ::; 2Iog(2+!xl),
Ir(x-y)I ::;~ Iog(2+lxl);w4
and if n = 2, applying (3.18) and (3.20),
Therefore, for n = 3 or n ~ 5,
(3.23) I3
(x ) ::; M(n)(1+lxl)4-nJ (l+lyl)-rdyIYI::;2+3Ixl
{
(1+lxl)4-r , if n ~ 5, r < n,
::; M(n,r) (1+lxl)4-n1og(2+lxl), if n ~ 5, r = n,
(1+lxl)4-n , if n = 3 or n ~ 5, r > n.
(3.25)
For n = 4,
For n = 2,
13(x)
s .!.(1+!xl)2 l 0g(2+\Xj)J (l+lyj}-rdyw lyl~2+3Ixl
~ M(r)(1+lxl)2 l og(2+lxl).
Now combining (3.10), (3.15), (3.17), (3.23)-(3.25), we obtain the
required estimate (3.4).
51
(b) In order to derive (3.5)-(3.7) we write
Zf(x) - ~or(x) = J 4[r(x-y) - r(x)]f(y)dy (n = 4),IR
Zf(x) - 1or(x) + 11.Dr(x) = J 3[r(x-y) - rex) + Dr(x).y]f(y)dy (n=3) ,IR
J 1 2 T= 1R2[r(x-y) - rex) + Dr(x).y - 2 D r(x).(yy )]f(y)dy (n = 2),
and apply (3.9) to obtain
(3 .26) IZf(x)-~l(x) I ~ II f l'-r [ 11(x)+1 2(x)+Jo(x)+Ko(x)+N1(x l l (n=4) ,
(3.27) IZ(x)-1or(x)+~1.Dr(x)1
~ II f ll_r [ 11 (x)+1 2(x) +J0 (x)+J1(x)+Ko(x) +K1(x)+N2(x) ] (n=3) ,
(3.28) IZf(x)-~or(x)+11.Dr(x)-12.to2r(x)1
~ II f lLr [ 11(x)+1 2(x)+Jo(x)+J1(x)+J2(x)+Ko(x)+K1(x)+K2(x)+N3(x)]
(n=2) ,
where 11(x), 12(x) are as defined in (3.11) and (3.12), and we futher
define
52
J. (x)1
K. (x)1
.- Inir<x)IJ lyli(1+lyp-rdy, (i = 0,1,2),ly-xl~(l+lxl)/2
.- Inif(x)IJ IYli(l+IYI)-rdY, (i = 0,1,2),ly-xl~2(1+lxl)
:= J If(x-y) - fCx)l(l+lyl)-rdy,(1+lxl)/2~ly-xl~2(1+lxl)
:= J IfCx-y)-f(x)+yenf(x)l(l+lyl)-rdy,(1+lxl)/2~ly-xl~2(1+lxl)
N3
(X) .-
J If(x-y)-f(x)+yenfCx)-io2f(x)eyyTIC1+lyl)-rdy.C1+lxl)/2~ly-xl~2C1+lxl)
By the definition of fCx) in (3.2), we have
(3.29) {-~ ,
nfCx) =8~ (2loglxl - 1),
if n = 3,
if n = 2;
and
loglxl[
l og [xI2 1
D I'(x) = 411"(3.30)
Ixl2
In JiCx), i = 0,1,2, we have (3.14); thus,
J.(x) ~ MCr)lnir<x)IC1+\xl)i-rJ . dy ,1 lyl~3(1+lxl)/2
which gives
{ (1+lxlj'-Tlxl'(lloglxll+l1, if n = 2,
(3.31) Jo(x) S; M(n,r) C1+lxl )3-r lx l , if n = 3,
C1+lxp4-rlloglxll , if n = 4;
{ (I+Ixll'-T [x] (lJogl xll+l) , if n = 2,(3.32) J
1(x)~ M(n,r) 4 r
(1+ Ix I) - if n = 3;
(3.33) J 2(x) ~ M(r)(1+lxl)4-r(lloglxl 1+1), when n = 2.
Applying (3.16) to Ki(x), i =0,1,2" we obtain
Ki(x) ~ Inif(x)IJ (~)i-rdYly-xl~2(1+lxl) 2
~ Inif(x)IJ 2nlzli-rdz,Izl~l+lxl
which yields
53
(3.34)
(3.35)
(3.36)
{
(1+lxl)2-rlxI2l0g(2+lxl), if n = 2,
Ko(x) ~ M(n,r) O+lxl)3-rlxl , if n = 3,
O+lxl )4-rlIoglxllif n = 4;
{
(1+lxl)3-rlxllog(2+lxl), if n = 2,KI(x) ~ M(n,r) 4
(l+lxl) -r if n = 3;
K2(x) ~ M(r)(1+!xl)4-r l og(2+lx\), when n = 2.
To estimate NI(x) (n = 4), we write, considering f as a function of
Ixl rather than x,
f(x-y) - f(x) = - 4~ t-I(lx_yl - Ixl),4
where t is some positive number detween Ixl and Ix-yl. Since (3.18)
holds in NI(x), we have
(3.37) Ixl ~ Ixl/2, Ix-yl ~ Ixl/2 =* t, Ix-syl ~ Ixl/2 for s E [0,1],
and hence
[I'(x-y) - rex) I ~ Mlxl-IIYI (n = 4).
Substituting into NI(x), again noting (3.18), we have
NI(x) S; Mlxl-IJ lyl(l+lyl)-rdy (n = 4),lyl~2+3Ixl
which gives
(3.38)
54
We substitute (3.15), (3.17), (3.31), (3.34), (3.38) into (3.26),
requiring that Ixl ~ 1, and obtain the desired estimate (3.5).
To estimate N2(x) (n = 3), and N3(x ) (n =2), we define
f(t) := T(x-ty), t E~.
By Taylor's Theorem,
f(l) - f(O) - f'(O) = ~ f"(s), for some s E [0,1],..and
f(l) - f(O) - f'(O) - ~ f"(O) = ~ fffl(s) , for some s E [0,1],2 6
and therefore, we have
(3.39) T(x-y) - rex) + nT(x)ey = - 8~ {Ix-yl - Ixl + lit}=-1~1I"{ lyl2 _[(x-sy)ey]2} (n=3),
Ix-sy I 1x-sy 13
and
into
loglxl
(x-sy)ey } (n = 2).Ix-syl2
by substituting (3.39)
1 2 T(3.40) r(x-y) - rex) + nr(x)ey - 2 D r(x)eyy
=.; {lx-YI'(]oglx-YI-ll-lxl'(]og\xl-ll+(2]OgIXI-1)XOY
21 Xl
[
loglxl - 2 +~
x1x2
Ixl2
= __1__ {2 [(X-Sy)ey]3 _ 31yl22411" Ix-syl4
Since (3.18) and (3.37) hold in N2(x),
N2(x), we obtain
N2(x) ~ Ml xl-1J lyI2(1+lyl)-rdy (n = 3),lyl:::;2+3Ixl
which gives
(3.41)
55
We substitute (3.15), (3.17), (3.31), (3.32), (3.34), (3.35), (3.41)
into (3.27), requiring that Ixl ~ 1, and obtain the desired estimate
(3.6).
Now substituting (3.40) into N3(x),
again noting (3.18) and
(3.37), we have
N3(x)
~ Mlxl-1J IYI3(1+IYI)-Tdy (n = 2),lyl~2+3Ixl
which gives
(3.42) N3(x)
~ M(T)lxl-1(1+\xl)5-T, if 4 < T < 5, n = 2.
Finally, we substitute (3.15), (3.17), (3.31)-(3.36), (3.42) into
(3.28), requiring that Ixl ~ 1, and obtain the desired estimate
(3.7). I
From the definition of rex) in (3.2), it is easy to derive the
following estimates for the first, second, third and fourth
derivatives of r:
I~. rex) I1
if n ~ 3,
if n = 2;
if n ~ 3,
if n = 2;
if n ~ 2.
\ax.g:.axk
r(x)1 ~ ~:3 \xI 1-n
, if n ~ 2;1 J n
laxiax~~kaxl rex) I ~By using the proofs of [11, Lemma 4.1, Lemma 4.2], the following
56
lemma concerning the Z-potential on bounded domains can be proved from
the above estimates.
LEMMA 3.2. Let f be bounded and locally Holder continuous (with
exponent 0 < a < 1) in a bounded domaim O. Then
(3.43) z(x) := J r(x-y)f(y)dyn
(3.44)
is in C4(0); and for any x E 0,
a~x) =J ar~-y) f(y)dy, p2 z(x) - J a2r(x-y)
f(y)dy,uA i 0 i OxiOx j - 0 OxiOx j
3 3a Z(x) _ J a r(x-y) f()dOx.Ox.Ox
k- n Ox.Ox.ax
ky y,
1 J u 1 J
a4z (x ) J a4r (x- y) )
Ox Ox Ox Ox = Ox ax a a [f(y -f'(x l Idyi j k I 0
0i j xk xl
( J a3r(x-y) )- f x) a a Ox "i (y dS.an x . x , k Y
o 1 J
Here i,j,k,l = 1,2, ... ,n, v = (VI' V2' ... , vn) is the outward
normal direction, 00
is any domain containing 0 for which the
divergence theorem holds, and f is extended to vanish outside n.
From Lemma 3.2 we have, for any x E n,
if n = 2,
if n ~ 3,
~Z(x) =J ~ r(x-y)f(y)dyn x
= { (2-~)WnJn -1-x~;;;"'y-l-n---2 f(y)dy,
__1__ Jloglx-ylf(y)dyW2 n
which is the well known Newtonian potential of f on the domain n.
(3.45)
Therefore, we also have, for x E n,
(3.46)
57
LEMMA 3.3. Let f be real valued on Rn and in some space B , with-r,o:
(3.47) a(zf) = Sf,
and
(3.48) ~2(Zf) = ~(Sf) = f
if n = 2
if n z 3,
where
(3.49 ){
(2-~)W J n Ix-yI2-nf(y)dy,n R
Sf'( x) :=
2~ JR
2 loglx-ylf(x)dy
is the well known Newtonian potential of f on Rn . Moreover,
. TOnIn Il'. ,
(a) if n ;:: 5, then Zf vanishes at infinity;
(b)'Yo
ifn = 4, then Zf + --4--loglxl vanishes at infinity;w4
'Yo 'Y eX(c) 1 . hif n = 3, then Zf + 81l"l x l - 8iTiT vanIS es at infinity;
(d) if n = 2, then Zf + ~1l" [ -'Yo lx I2( l og lx l- 1) + (210glx!-1) xe'Y
1
Xe'Y x ]- (loglxl - +)tr('Y2) - Ix~2 vanishes at infinity.
Remark Being aware of the fact that Dir(x) vanishes at infinity when
n ;:: 5-i, (i = 0,1,2), we can write (a)-(d) of Lemma 3.3 in the
following uniform way: for n ;:: 2,
(3.50)
Proof of Lemma 3.3: By choosing r smaller if necessary, we may
prescribe 4 < r < n if n ;:: 5, and 4 < r < 5 if n = 2, 3, 4. Then
(a)-(d) follow from Lemma 3.1.
58
In order to discuss differentiability of Zf, we let p denote
some large radius and form the decomposition
if n ~ 3,
Zf(x) =J r(x-y)f(y)dy + J r(x-y)f(y)dyIyl~p Iyl~p
= Z (x) + w (x)p p
Lemma 3.2 gives that zp(x), the first integral of (3.51), is 04 in the
region 0p = {xElRn : IAI < p}, with (3.45), (3.46) valid if we replace
z(x) and °by zp(x) and 0p' It is clear that wp(x) , the second
integral of (3.51), is COO in 0p with
{
JIYI~p Ix_yI2-n(1+lyl)-Tdy,
(3.52) IAWp(X) I s M(n)lI f ILT
J Iloglx-yll(l+lyl)-Tdy, if n = 2,Iyl~p
(3.51)
and
in np
From (3.52), it follows easily
(3.53) a2wp
(x ) == 0
Since p is arbitrary, Zf is 04 in Rn.
that aw (x) ~ 0, ~s p ~ 00, for any (fixed) x ERn. By lettingp
p ~ 00 in (3.52)-(3.53), and also in (3.45)-(3.46) with z(x) and n
replaced by zp(x) and 0p' we obtain (3.47) and (3.48). I
Douglis and Nirenberg have proved a Liouville-type theorem for
certain homogeneous elliptic systems with constant coefficients and
homogeneous order (Theorem 3, [4]). Applying their result to a single
equation of fourth order, we obtain the following:
LEMMA 3.4. (Liouville-Type Theorem) Let ~ be a 04 solution of
a.D4~ = 0 in Rn,
where a is a constant tensor in ~nxnxnxn and satisfies the elliptic
condition (2.8), and assume that, for some nonnegative integer m,
~(x) = o(lxl m) at infinity.
Then ~ =0 if m = 0, and ~ is a polynomial of degree ~ m-1 if m > O.
Particularly, the result holds for entire solutions of the biharmonic
equation (in which case, D4 = ~2).
The following basic result for the nonhomogeneous biharmonic
equation will serve as our model for the general fourth order elliptic
equation.
59
THEOREM 3.5. Let f be real valued on ~n and in some space B with-T,O:
T > 4, 0 < 0: < 1. Then there exists exactly one entire C4 solution ~
of the equation ~2~ = f, namely the solution ~ = Zf, such that
(a) when n ~ 5, ~ vanishes at infinity; if moreover 4 < T < n,
then for this solution we have the bound
(3.54) 1IC/>1I 4_T,1 + 1IDcf>11 3_T, 1 + IID2C/>11
2_T, 1 + IID3~1I1_T, 1 + liD ~II_T, a
~ M(n, a, T)lIfll-T,a
(b) when n = 4, ~(x) - 1or(x) vanishes at infinity for some
constant 10
; moreover, for this solution c/> and any positive
constant € we have the bound
(3.55) 11c/>II E,1 + IIDtI>II E_1,1 + Iln2~IIE_2,1 + IID3~11€_3,1 + IID4~IIE_4,a
s M(a, T, dllfll ;-T,a
(c) when n = 3, c/>(x) - 1or(x) + 11enr(x) vanishes at infinity for
some constant 10
and constant vector 11; moreover, for this
solution we have the bound
(3.56) 114>11 1,1 + 1ID4>1I0,1 + IID24>11_1, 1 + IID34> 1I _2 , 1 + IID44>1I_3,a
:;;; M(a, r)lIfll ;-r,a
(d) when n = 2, 4>(x) - 10r(x) + 11eDr(x) - t 12eD2r(x) vanishes at
infinity for some constant 10, constant vector 11 and constant
symmetric matrix 1 2; moreover, for this solution 4> and any
positive constant E we have the bound
60
(3.57) 114>II E+2,1 + 11D4>II E+1, 1 + IID24>II
E , 1 + IID34>IIE_ 1, 1 + IID44>II
E_ 2,a
:;;;M(a, r , E)llfll .-r,a
The uniquely determined constant 10, constant vector 11 and constant
matrix 12 in (b), (c) and (d) are defined by (3.8).
Proof: (a) By Lemma 3.4 for biharmonic functions (also see [6, 8]),
uniqueness of the solution is clear since the difference of two
solutions would be biharmonic and vanish at infinity, thereby being
identically zero. Lemma 3.3 states that 4> = Zf is a solution
vanishing at infinity. In order to derive the bound on 4>, we apply
(2.14) of Theorem 2.4 to the special case of the biharmonic operator,
with u = 4 - r, obtaining
114>1I 4_r,1 + I1D4>1I 3-r ,1 + IID24>112_r, 1 + IID
34>111_r, 1 + IID44>II_
r ,a
s M(n, a, r) {1I4>11 4_r + II fll_r ,a}'
But Lemma 3.1 implies, for 4 < r < n,
which, upon substitution into the previous inequality, yields (3.54).
(b) For uniqueness, suppose there were two C4 solutions 4>1' 4>2
and real constants 101, 102, with 4>i(x) - 10ir(x) vanishing at
61
infinity, i = 1, 2. Setting ~ = ~1 - ¢2' 10 =101- 1 02, we have
2 1 0~ ~ = 0, and ~(x) + 4w loglxl vanishes at infinity. By Lemma 3.4,
4
~ =constant, and therefore 10 = 0 and ~ =O.
Again, Lemma 3.3 states that ~ = Zf is a solution of the problem,
with 10 defined by (3.8). To obtain the bound on~, we observe that
since
(3.58)
Lemma 3.1 (a) implies the inequality
We also have, since £ - 4 > -r,
Il f ll £_4 a s Ilfll_r 0:', ,
We substitute these inequalities into the right-hand side of (2.14),
as applied to the biharmonic equation with u = E, and obtain (3.55).
(c) For uniqueness, we need to prove that the only solution of
~2~ = 0 on R3, with
1 0 1 1 - x 3(3.59) ~(x)+8~lxl~vanishing at infinity for some 10E R, 11E R ,
is ~ =0, in which case 10 = 0, 1 1 = O.
By Lemma 3.4, we have ~(x) = q_x + p, a polynomial with degree
::;; 1. If 0: ~ 0, we choose a unit vector i such that q-i = 0; then
setting x = ki in (3.59) and letting k ~ 00, we conclude that 10 = 0,
and then q = O. Therefore, we must have ~ =p, 11 = 0 and
(3.60)
Now, if 11
in (3.60),
1 1 -X
P - 81iTiT ~ 0 as x ~ 00.
~ 0, by letting x = k11, k ~ 00, and then x = -k11, k ~ 00
we get p = ~~1111 =- ~~1111. Thus we have proved that
62
¢ = 0, 10 = 0 and 11 = O.
Again, Lemma 3.3 states that ¢ = Zf is a solution of the problem,
with 10 and 11 defined by (3.8). The bound (3.56) follows from (2.14)
of Theorem 2.4 (with u = 1), Lemma 3.1 (a) and Lemma 2.3 (b).
(d) For uniqueness, we must show that the only solution of
2 2/),. ¢ = 0 on IR ,
(3.61) 8~¢(x) - 10IxI2(10glxl-1)
21 Xl
[
loglxl - 2 +~
- 12 -x
1x2
Ixl 2 loglxl 1-2+
~ 0,
where
(3.62)
~ - f 0 ~ EIRE 1R2 symmetrl'c ~2 E 1R2X2,as x --~, or s me '0 ,1 11 ,
is ¢ =0, in which case 10 = 0, 11 = 0 and 12 = O.
By Lemma 3.4, ¢ is a polynomial of degree ~ 2. Therefore, from
(3.61), we must have 10 = O. Thus ¢ = 0(lxI 2) at infinity, and again
Lemma 3.4 implies that ¢ is a polynomial with degree ~ 1. Hence,
again from (3.61) (with now 10 =0), 11 = O. Applying Lemma 3.4
again, we conclude that ¢ =p, a constant, and (3.61) becomes
2
8~ - (loglx I _1. + ~.!_J'"1 - (lOg [xI2 Ix 12) 11
2x1x2- -- 1 ~ 0 as X ~ 00,I X 12 12 '
(1 11 112] := '"1
2112 122
From (3.62), we must have
(3.63)
Now by letting x
we get
(3.64)
and
T= (xl' 0) -4 00 and then X =
63
1 1(3.65) 8~P + 2 1 1 1 - 2 '22 = o.
Solving the linear system (3.63)-(3.65), we obtain ~ =p = 0, 'II =
'22 = 0; then, from (3.62), we also have '12 = O. Thus we have proved
that ~ = 0, , = 0, , = 0 and, = O.o 1 2
Again, Lemma 3.3 states that ~ =Zf is a solution of the problem,
with '0' 'I and '2 defined by (3.8). The bound (3.57) follows from
(2.14) (with u = € + 2), (3.58), Lemma 3.1 (a) and Lemma 2.3 (b). I
4. AN A PRIORI BOUND FOR THE FOURTH ORDER ELLIPTIC EQUATION
We consider the nonhomogeneous equation,
(4.1)
and derive an a priori bound for entire solutions analogous to
Theorem 3.4. The conditions on the coefficients will guarantee that
the operator L approaches the biharmonic operator near infinity at a
certain rate; and the condition of the separability of L (= L2L1
)
will be used to prove the uniqueness of the solution. Let J denote
the nxnxnxn tensor (OijOkl) with J.D4 = a2• We require:
Condition F (i) The operator L can be separated as L = L2L 1
, where
L = a .D2+ b .D + c (s = 1, 2),s s s s
and a , b , c are matrix, vector, scalar valued functions on Rn,s s s
respectively, with aI' bl and· c1
being C2 functions and
(4.2) Cs ~ 0 for n ~ 3, and Cs =0 for n = 2 (s = 1, 2).
For these coefficients, there exist constants 8, a, A, with
°> max{O, 4-n} , 0 < a < 1, A ~ 0, such that
(4.3) lias - IILo a' IIbsll_1 _O a' Il cs ll _2_0 a', , ,
IIDsaIILs_8,a' IIDsbll1_l_s_o,a' IIDsclll_2_s_0,a::;; A, s = 1, 2.
(ii) The matrices as' s = 1, 2, are symmetic, and for some
constant A, A > 0,
64
Remark
(4.4)
From L = L2L1, i.e., from
43222a-D +b.D +c-D +d.D+e = (a 2-D +b2eD+c2)(al eD +bl·D+c1),
and from Lemma 2.3, it follows from condition (4.3) that
(4.3)' lIa - JII_o,a' Ilbll_I _8,a' II cll_2_0,a' IIdll_3_8,a' lIell_4_0,a s M(A).
So, under Condition F, we can use both (4.3) and (4.3)'.
THEOREM 4.1. Suppose Condition F holds, that fEB for some T,-T,a
T > 4, and that a is the same as in Condition F. Let ~ be an entire
C4 solution of (4.1).
(a) For n ~ 5, assume that ~(x) vanishes at infinity. Then
(4.5) 114>11 0 , 1 + IIDt1>II_ 1 , 1 + IID2~11_2,1 + IID3~IL3,1 + IID4~1I_4,a
~ M(n, a, 8, T, A, A)llfll .-r,a
Moreover, for any c such that, € ~ min {o, T-4},
(4.6) lIa24>11_
4 _ € ,a ~ M(a,8, T ,A,A) II fll_T ,a'
and when also 0 < € < 1 and Ixl ~ 1,
(4.7)
65
(b) For n = 4, assume that ~Cx) - ~ologlxl vanishes at infinity
for some constant ~o. Then
(4.8)
and for any E > 0,
C4.9) 1I~IIE,I+ 1ID<PII E_1,1 + IID2~IIE_2,1 + IID3~IIE_3,1 + IID4~IIE_4,a
::; M(a,o,T,E,A,A)lIfll .-T,a
If, moreover, E < 0, E ~ 1"-4, then
C4.10) lIa2~IL4_E a::; MCa,o,T,E,A,A)lI fILT a', ,and if also 0 < E < 1 and Ixl ~ 1, then
C4.11) I~(x)-~ologlxll::; M(a,o,r.E,A,A)llfll_T a(1+lxP-Cl ogC2+lxIL,~lex
Cc) For n = 3, assume that ~(x) - ~olxl +~ vanishes at
ninfinity for some ~o E ~ and ~1 E ~. Then
(4.12) I~ol, h 11 s MCa,o,T,A,A)llfll_T a',
C4.13) 114>11 1,1 + 11D<Pllo,1 + IID2~1I_1,1 + IID34>1I_
2,1 + IID4~1I_3,a
::; MCa,o,1",A,A)llfll .-1",a
Moreover, for any c such that E ::; min {a-I, 1"-4},
C4.14) 11L\24>11_4
_ E a ::; M(a, a, 1", A,A) Il f ll _1" a', ,and when also 0 < E < 1 and Ixl ~ 1,
C4.15) I~Cx) - 'Yolxl + ~le61 ::; MCa,a,1",E,A,A)lIfll Cl+lxl)-E.IXI -1",a
loglxl
(21og!xl-l)'Yl ex
2I Xl
--+--2 [x1
2
Cd) For n = 2, assume that
~Cx) - ~olxI2(IOglxl-l) +
- ._. _._- ----------
vanishes at infinity for some constant 10
, constant vector 11
and
constant matrix 12, Then
66
(4.16)
loglxl
and for any E > 0,
(4 .17) 11~IIE+2,l + IIDt1>II E+1,l + IID2~IIE,l + I\D3~IIE_1, 1 + IID4~I\E_2,0:
:;; M(o:,o,r,E,>',A)llfll .-1",0:If, moreover, E < 0-2, E ~ r-4, then
(4.18) l\a2~IL4_E,0: ~ M(o:,O, r , E,>',A)lIfll_r ,0:'
and if also 0 < E < 1 and Ixl ~ 1, then
(4.19) I~(X) - 1o lx \2( l og lx l- 1) + (2Ioglxl-1)11•x
21 xl
- -+--2 Ixl2
Ixl2
~ M(o:,o,r,E,>',A)lIfll C1+lxl)-E l og(2+lxl).-r,o:
Proof: (a) First assume that (4.5) holds for all C4 solutions ~ such
th&t ~ vanishes at infinity.
(4.20)
where
We rewrite (4.1) as
a2~ = g,
(4.21) g := -(a-J).D ~ - b.D if> - c.D if> - d.D¢ - e~ + f.
If E is any fixed real number with E :;; min{o, r-4}, then from (4.20),
(4.21), Lemma 2.3 (b) and (c), Condition F (with (4.3)'), and (4.5) we
have the bound
(4.22) lIa 2if>1I_4_E,0: s Ila-JII_E,o:IID4if>1I_
4 ,0: + IIbIL1_E,0:1ID3~1'-3,0:
+ I\ cll_2_ E,o:I\D2if>11_2,0: + Ildl'-3_E ,0:1ID¢1'-1 ,0:
"..,u.
+ lIell_4 _ £ a ll4>ll o a + II fll_4 _ £ a" ,
s M(AHIID44>11_
4 , 0: + IID34>IL
s , a + IID24>IL
2, 0:
+ 1ID4>IL 1 0: + 114>11 0 a} + II fLr 0:, , ,~ M(a,o,r,A,A)lIfll ,-r,a
which is (4.6). When also 0 < £ < 1 and Ixl ~ 1, the inequality (4.7)
is a consequence of Lemma 3.1 (a) and (4.22).
Therefore, in order to finish the proof of (a) it is sufficient
to show that (4.5) holds. We choose £ sufficinetly small that
(4.23) 0<£ < min {1, 0/2, r-4}.
Observe that (4.3) implies
(4.24)
lIas ll o,o: ~ Ilas-Illo,o: + 11 1110 , 0: ~ Ilas-III_o,a + 11 111 0 s A + vn,
IIbs ll _1 a ~ IIbsll_1 _0 0: ~ A, II csll_2 a ~ II cs ll _2_0 0: ~ A,, , , ,
IIDsallLs 0: ~ IIDsalll_s_o 0: s A,, ,
IIDsblll_l_s a s IIDsblll_l_s_o 0: ~ A,, ,
IIDscllI_2_s a ~ IIDscllI_2_s_0 0: ~ A (s = 1, 2)., ,Hence, from (4.4) and Lemma 2.3, we have (2.13) with A replaced by
M(A). Thus Theorem 2.4 can be applied with u = 0 to get the estimate
(4.25) 114>11 0 1 + 1ID4>1I_1 1 + IID2¢11_2 1 + IID34>11_
3 1 + IID44>IL4 a, , , , ,
~ M(n,a,r ,A,AHII4>lI o + Ilfll_r a},,Therefore it is sufficient to derive, instead of (4.5), the more
conservative estimate
(4.26) 1I¢llo ~ M(n,o:, 0, r ,A,A)Ilfll_r a',Suppose that (4.26) is false. Then there exist sequences {a },sm
{bsm}, {csm}' {fm}, {4>m}' s = 1, 2; m = 1, 2, 3, '" , such that all
a ,b ,c satisfy Condition F with the same constants 0, 0:, A, A,sm sm sm
such that each f is in B ,each ~m is of class C4 and vanishes atm -1' ,a:
infinity, with
68
(4.27)
and with
3+ b .D ~m m
+ b .D +2m
2+ c .D ~ + d.~ + e ~m m m m mm
c2
)(al
.D2~ + bl.~ +m m m m m
(4.28) lI~ml\o = 1, I\ fmll_T , a: s ~ .Then inequalities (4.24), (4.25) are satisfied by the members of
these sequences; and in particular, for s = 1, 2, the sequences
{asm-I}, {bsm}' {csm}' {Dsa l m} {D
sblm}, {D
sclm}, {fm}' {~m}' {~m}'
{D2~m}' {D3~m}' {D4~m} (s =1,2) are all uniformly bounded with
respect to the appropriate norms in the spaces B r ,B r ,-U,a: -1-u,a:
B c ,B s: ,B 1 r ,B 0 ".J; ,B._", r•.' Bo, l' B_1 , i ' B_2, 1'-2-u,a: -s-u,a: - -s-u,a: -M-U-V,a '.
B B respectively. By Lemma 2.3 (f), there exist functions-3,1' -4,a:'
as-I, bs' cs' f, ~ together with DSal, DSb
l, DScl (s = 1,2), ~, D2~,
D3~, D4~ all contained in the same spaces as the corresponding
sequences such that (noting (4.23)), after passing to subsequences
(preserving (4.28)), we have with respect to the norms of the spaces
involved,
(a -I) ~ (a -1) in B ,and hence a ~ a in B ,sm s -E sm s -E
in B-2-c'
~ DSb in B1 -1-S-E'
(4.29)
b ~ b in B c ~ csm s -1-E' sm sDSa ~ DSa in B ,DSb
rm 1 -S-E i mDSc ~ DSc in B (s =
1m 1 -2-S-E1, 2),
f ~ f in B 4' ~ ~ ~ in B ,m - m E
~ ~~ in B l' D2~ ~ D2~ in BE-2 'm E- m
D3~ ~ D3~ in B 3' D4~ ~ D4~ in BE-4'm E- m
and, from Lemma 2.3, we also have
69
(4.30)
From (4.28) we obtain
IIf ll _4 = lim Ilfmll_4 s lim IIfmll_r s lim (11m) = O.
Therefore, upon passing to the limit in (4.27) we find that ~ is an
entire C4 solution of the homogeneous equation
(4.31) aeD4~ + beD3~ + ceD2~ + deD~ + e~
2 2= (a2eD + b2eD + c2)(al eD ~ + b1·D4>
+ cl~) = O.
Moreover, it is clear that (4.2) and (ii) of Condition F on c and as s
2are fulfilled. Since (4.30) implies that ~ = aleD ~ + ble~ + cl~
vanishes at infinity, the maximum and minimum principles (applied to
arbitrarily large spheres in ~n) for second-order elliptic equations
([11], [17]) may be applied to solutions ~ of the equation (4.31) to
obtain
(4.32)
We need to show that ~ =0, but first we have to investigate the
bahaviour of each ~m near infinity. We again rewrite (4.27) as
(4.33)
where
(4.34)
2(). 4> = g ,m m
g := -(a -J)eD44> - b eD34> + c eD24> + d enq, + e 4> + f .m m m m m m m m m mm m
Then, from Lemma 3.1, Condition F (with (4.3)') and (4.28) for am' bm,
(4.35)
e and 4> , f , we have the estimatem m m
Ilgmll_ 4_€.a ~ II am-JIL2€ ,aIlD4~mll€_4,a + IlbmIL l _ 2f: ,aIID3~mll€_3 ,a
+ IlcmL 2_2€ allD2cPmllc_2 a + IIdmll_3_ 2€ a llD4>mll€-l a, , , ,
+ lI emL4_ 2f: all~mllE a + I fm"_4_t a" ,
(4.37)
70
~ M(A){M(a,€,~,A).2} + 1.
Thus g E B 4 ' and then Theorem 3.5 (a) and Lemma 3.1 (a) implym - -€,a
(4.36) 4Jm = Zgm' l4Jm(x)I s M(n,dllgmll_4_€(l+l xl)-€.
By substituting (4.35) and letting m -4 00, we see that 4J vanishes at
infinity; and then by applying the maximum and minimum principles (to
(4.32», we conclude 4J =O. Hence (4.30) yields ~ -4 0, ~ -4 0,m m
D2~m -4 0, D3~m ~ 0, D4~m ~ 0 in the spaces B , B , B , B ,€ €-1 €-2 €-3
B , respectively. Again from (4.34), (4.3)' and (4.28) for am' bm,€-4
e and ~ , f , we observe thatm m m
IIgmll_4_€ s II am-JII_ uIlD\Pmll€_4 + IlbmLl_uIlD34Jmll£_3
+ IlcmIL2_uIlD24Jmll£_2 + IIdmIL3_ullD4Jmll€_1
+ lIemIL4_ull4Jmll€ + II fmll_r
~ M(AHIID44>m
ll€_4 + IID34>mll£_3 + IID24>mll£_2+ IID4>mll£_1 + l14>mll£} + ;.
Hence Ilg II ~ 0 as m~ 00. But then from (4.36), Lemma 3.1 (a) wem -r
obtain
l14Jmll£ ~ M(n,e) IlgmIL4_£ ~ 0 as x ~ 00,
which contradicts the assumption (4.28).
(b) First assume that (4.9) holds for all £ > 0 and all C4
solutions 4> such that 4>(x) - 1ologlxl vanishes at infinity for some
constant 10
, If € < 8, £ ~ r-4, then from (4.1), Condition F (with
(4.3)'), and (4.9) (with € replaced by 8-£) we obtain the bound
(4.38) 1IL\24J11_4_£ a,
s Ila-JII_8,aIlD44>118_€_4,a + Ilbll_I_8,aIlD34>118_£_3,a
---------------- ------------------------------
+ Il cll_2_o, aIID2if>lIo_E_2, a + Ildll_3_o, allDif>llo-E-1 , a
+ lI e ll_4_o allif>lIo_E a + II f ll_4_E a" ,
s M(AHIID4if>lIo_c_4 a + IID3if>lIo_c_3 1 + IID2if>llo_c_2 1, , ,
+ 11Dif>llo_E_1 , 1 + 1Iif>lIo_c,1] + II fll_r, a
~ M(a,o,r,c,A,A)lIfli ,-r,a
which is (4.10). Moreover, by Theorem 3.5 (b),
(4.39) if> = Z(~2if», 10
= - 4~ J 4(~2if»(Y)dY,4 IR
which gives the bound
(4.40) 110 1 s ~4 JIR411L12if>1I_4_c(l+lyl )-4-Edy s M(c)IIL12if>1I_4 _ c'
We take c = min {O/2, r-4} (thereby removing dependence on c), and
combine (4.40) with (4.38) to obtain (4.8). When also 0 < c < 1 and
Ixl ~ 1, the inequality (4.11) is a consequence of Lemma 3.1 (b) (with
r replaced by 4 + c) and (4.38).
Therefore, in order to complete the proof of (b) it is sufficient
to show that (4.7) holds for all c > O. First we observe that the
left-hand side of (4.7) increases as c decreases. Thus we may take c
smaller, if necessary, so that
71
(4.41) o < c < min {I, O/2}, E ~ r - 4.
The inequalities (4.24) hold as when n ~ 5, and if if>(x) - 1010glxl
vanishes at infinity for some 10, then if> E~c for all c > O. Thus we
may apply Theorem 2.4 (with u = c) and conclude, analogously to
(4.25),
(4.42) 11if>ll c, 1 + 11Dif>ll c_ 1 , 1 + IID2if>llc_2, 1 + IID3if>llc_3,1 + IID4if>llc_4 ,a
~ M(a, e ,A, A){ 11if>llc + II f ll_r a}',
72
Therefore it is sufficient to derive the estimate
(4.43) II¢II s M(o:,O,1",E,A,A)lIfll .E -1",0:
Assuming that (4.43) is false, we obtain sequences {asm}' {bsm}'
{csm}, {fm}, {¢m}, born}, s = 1, 2; m = 1, 2, 3, '" , such that all
a , b , c satisfy Condition F, such that each f is in B , eachsm sm sm m -1",0:
¢ is of class C4 and a solution of (4.27), with ¢ (x) - ~o loglxlm m m
vanishing at infinity, and with
(4.44)
We substitute (4.44) into (4.42)
obtain
II f mll_1" 0: ~ ; .,(applied to all pairs {¢m,fm}) and
(4.45) lIc/>mllE 1 + lID4>mIlE_1 1 + IID2¢mIIE_2 1 + IID3¢mIIE_3 1 + IID4¢mIIE_4 a, , , , ,~ M(0:,O,1",E,A,A)(2).
Since °> °- E > E > 0 (see (4.41» we may pass to subsequences so
that
a -I ~ a -I in B ~, b ~ b in B ~ l'sm s E-u sm s E-u-
C ~ c in B ~ 2' DSa
l ~ DSal
in B ~ ,sm s E-u- m E-u-S
DSb ~ DSb in B ~ , DSc ~ DSc in B ~ (s=1,2),1m 1 E-u-S-l 1m 1 E-u-S-2
(4.46) fm~ 0 in B_4, ¢m ~ ¢ in BO_E
'
D4> ~ D4> in B~ i ' D2¢ ~ D
2¢in B~ 2'm u-E- m u-E-
D3¢ ~ D
3¢ in B~ 3' D4¢~ D
4¢in B~ 4'm u-E- m u-E-
Then ¢ is an entire C4 solution of the equation (4.31) and, from
Lemma 2.3 (f) and (c),
(4.47) ~ := a eD2
c/> + b eDt/> + clc/> E B 2'1 1 E-
We will conclude that c/> =0, but first we must investigate more
carefully the behaviour of each ¢ near infinity. We again havem
(4.33), with gm defined by (4.34). From (4.34), (4.44), (4.45) we
have the estimate (4.35). Thus gm E B_4
_E
a' and Theorem 3.5 (b),
73
implies
4>. = Zg ,m m
By making estimates as in
10m = - 4~ J4 gm(y)dy.4 IR
(4.40), with (4.35) being applied, we see
that the sequence {1 } is a bounded sequence of real numbers. Onom
passing again to subsequences, we may assume 10m~ 10 for some real
number 10' Lemma 3.1 (b) implies that, for Ixl ~ 1,
Ic/>m(x) - 10mloglxll s M(E)llgmL4_/1+l xl )-E l og(2+lxl).
By substituting (4.35) and letting m~ 00, we obtain for Ixl ~ 1,
Ic/>(x) - 10loglxll ~ M(a,E,A,A)(1+lxl)-El og(2+lxl),
and hence
(4.48) c/>(x) - 10l0glxl vanishes at infinity, for some 10 E IR.
Since (4.47) implies that f vanishes at infinity, the maximum
and minimum principles for second order elliptic equations may be
applied to solutions f of the equation (4.31) to obtain
(4.49)
Since c/> E BE is a solution of (4.49), a modified theorem of Begehr and
Hile ([2], Theorem 5.3 with B replaced by B , E < min {1,o}) may bem m+E
applied to solutions of the equation (4.49) to get
c/>(x) - p ~ ° as x ~ 00 for some constant p;
but then (4.40) implies that 1 = p = 0, and thus c/> vanishes ato
infinity. Applying the maximum and minimum principles again, we
conclude that c/> =O. Hence (4.46) yields c/> ~ 0, Dc/> ~ 0, D24>. ~m m m
0, D3c/>m ~ 0, D4c/>m ~ 0 in the spaces BO_E
' BO_E_ 1' BO_
E_
2' BO_
E_
3'
B~ ,respectively. Again from (4.34), (4.44), we have an estimate0-£-4
(4.50) IIgmIL4-£ ~ \lam-JIL6I1D44>mIl6_c_4 + Ilbmll_I_61ID34>mIl6_c_3
+ Ilcmll_2_6I1D24>mIl6_£_2 + \Idmll_3_611D4>mI16_c_l
+ lIemll_4_6114>mI16_c + II fmIL4_£
~ M(A)[ IID44>mllo_c_4 + IID3
if>m11o-c-3 + IID24>mllo_c_2
+ lID4>mIl6_c_1 + l14>m11o_cl + ; .
Hence Ilg II ~ 0 as m~ 00. Finally, from Lemma 3.1 (a) we havem -4-c4for x E IR ,
l4>m(x) I ~ M(c)lIgmll_4_clog(2+lxl),
which implies that 1Ir/J II ~ 0 as m~ 00, contradicting (4.44).m c
(c) Again we first assume that (4.13) holds for all C4 solutions
4> such that 4>(x) - ~olxl + ~1.TiT vanishes at infinity for some
constant ~o and constant vector ~1' If c is any fixed real number
with c ~ min{6-1, r-4} , then from (4.1), Condition F (with (4.3)'),
and (4.11) we obtain the bound
(4.51) liD?4>11_4_c a s lIa- JII_1_c a11D24>1'-3 a + IIb ll_2_c aIID34>11_2 a, " , ,+ Il clL3_c aIID24>II_l a + Ildll_4_E a llD4>lIo a, , "+ lI e ll_5 _ E a ll4>11 1 a + II fll_ 4_c a
" ,~ M(A) [IID44>1I_
3 a + IID34> 1I _2 a + IID24>11_1 a, , ,
+ 1ID4>llo a + 114>11 1 a l + II fll_ r a, , ,:;:; M(a,O,r,A,A)llfll ,-r,a
which is (4.14). Moreover, by Theorem 3.5 (c),
2 1f 2 1f 24> = Z(~ 4», ~o = - 8~ 3(~ 4»(y)dy, ~1 = - 8~ 3 y(~ 4»(y)dy.IR IR
Thus, by (4.51), we have the bound
74
(4.52) I'Ysl::; 8;' JIR3IyIS(l+lyP-4-Ellil2C/>11_4_EdY ~ M(E)lIil2C/>1I_4_E
s M(E)lIil2C/>IL4
_ E a::; M(a,6,1",E,A,A)lI fIL T a (s = 0, 1)., ,Choose E = min {6-1, T-4}. Then M in (4.52) is independent of E, and
(4.12) holds. When also 0 < E < 1 and Ixl ~ 1, the inequality (4.15)
is a consequence of Lemma 3.1 (b) and (4.51).
Therefore, in order to finish the proof of (c) it is sufficient
to show that (4.13) holds. We choose E sufficiently small that
75
(4.53) o < E < min {1, (6-1)/2, T-4}.
Since inequalities (4.24) hold as when n ~ 5, Theorem 2.4 (with u = 1)
may be applied to obtain an estimate
(4.54) 1Ic/>11 1 1 + 1IDc/>lIo 1 + IID2c/>1I _1 1 + IID3c/>1I_2 1 + IID4c/>IL 3 a, , , , ,
s M(a,6,T,>',AHIIC/>11 1 + IIf lLT a}·,Hence it is sufficient to derive the estimate
(4.55)
Suppose that (4.55) is false. Then there exist sequences {asm}'
{bsm}' {csm}' {fm}, {c/>m} , {~om}' {~lm}' s = 1, 2; m = 1, 2, 3,
such that all a , b , c satisfy Condition F, each f is insm sm sm m
B , and each c/> is of class C4 and a solution of (4.27), with-1",a m
c/>m(x) - ~omlxl + ~1m.TiT vanishing at infinity and
(4.56) lIc/>mlll = 1, II f mll_r a:;;; ~ .,
Substituting (4.56) into (4.54) (applied to all pairs {c/> ,f }), we getm m
(4.57) Ilc/>mlll,l + IIDc/>m11o,1 + IID2c/>mll_1,1 + IID
3c/>mll_2,1 + IID4c/>mll_3,a
::; M(a,6,T,>',A)(2).
By (4.53), we may pass to subsequences so that
(4.58)
a -I ~ a -I in B ,b ~ b in B_2
_E
,sm s -1-E sm s
csm ~ Cs in B_3
_E
, DSaim~ DSa
l in B_ I_S_E'
DSb ~ DSb in B , DSc ~ DSc in B ,rm 1 -2-S-E im 1 -3-S-E
f ~ 0 in B , ~ ~ ~ in B l'm -4 m E+
~m ~ ~ in Be D2~m ~ D2~ in BE_I'
D3~m ~ D3~ in BE
_2
, D4~m ~ D4~ in BE
_3
(s = 1, 2).
Therefore, ~ is a C4 solution of the equation (4.31), with
2(4.59) ~ = a l-D ~ + bl-~ + cl~ E B_1,
We need to show that ~ =O. We again have (4.33) and (4,34),
and from (4.34), (4.56), (4.57), we also have, analogously to (4.51),
the estimate
76
(4.60) Ilgm1L4_E ex ::; M(A) [IID4~mll_3 a + IID3~mIL2 a + IID2~mll_1 a, , , ,
+ II~mllo a + II~mlll a] + IlfmlLr a, , ,::; M(A){M(a,8,r,A,A)_2} + 1.
Thus g E B , and then Theorem 3.5 (c) impliesm -4-E,a
(4.61) ~m = Zgm' 'om = - 8; S~3 gm(y)dy, 'Im = - 8; S~3 ygm(y)dy.
By making estimates as in (4.52) (for, and g ), with (4.60) beingsm m
applied, we find that the sequences {'om}' {'lm} are bounded sequences
of real numbers and real vectors, respectively. On passing again to
subsequences, we may assume 'om ~ '0 and 'lm ~'1 for some ' 0 E R
and '1 E ~3. Lemma 3.1 (b) implies that, for Ixl ~ 1,
I~m(x) - lom1x l + 11m-TiTI s M(E)llgmll_4_/1+lxl)-E.
By substituting (4.60) and letting m~ 00, we see that
(4.62) ~(x) - 'olxl + 'I-TiT vanishes at infinity,
3for some 'oE ~, 'IE ~ .
Since, from (4.59), 0/ vanishes at infinity, ihe-maximum and
77
minimum principles for solutions ~ of the second order equation (4.31)
imply that
(4.63) 0/ = al.n2~ + bl.~ + cl~ =0 in R3.
Again, since ~ E B1
is a solution of (4.63), Theorem 5.3 of [2] states
that
(4.64) ~(x) - qex - p ~ 0 as x ~ 003for some pER and q E R .
Then, (4.62) and (4.64) imply that
(4.65) qex + p - ~olxl + ~leTiT vanishes at infinity,
3for some p, ~o E R and some q, ~I E R. A proof analogous to that of
Theorem 3.4 (c) may be appied to ensure that p = ~o= 0 and q = ~I= O.
Hence ~ vanishes at infinity, and then ~ =0 follows form the maximum
and minimum principles (for the equation (4.63».
We now have from (4.58) that ¢ ~ 0, D¢ ~ 0, n2¢ ~ 0,m m m
D3¢ ~ 0, D4¢ ~ 0 in the spaces B ,B, B I' B 2' B 3'm m c+1 c c- c- c-
respectively. Again from (4.34), (4.56), we have
IIgmlL4- c s II am-JII_I_uIlD4c/>mllc_3 + IIbmll_2_2E1ID3~mllc_2
+ IIcmll_3_2EIID2c/>mllc_1 + Ildmll_4_2E11D4>mllc
+ lIemll-5_2cll~mllc+1 + Il f ll_4_c
::; M(A) [IID4~mllc_3 + IID3~mllc_2 + IID2c/>m
ll€_11
+ IIThPmll c + lIc/>mllc+l] + m'Hence IIgmll_4-€ ~ 0 as x ~ 00; and then, from Lemma 3.1 (a),
Ilc/>mlli s M( c) IlgmlL4_c ~ 0 as m ---? 00
which contradicts (4.56).
(d) As in the proof of (b), we first assume that (4.17) holds for
all £ > 0 and all C4 solutions ~ such that
(4.66)
loglxl
as x ~ 00, for some 70
E R, 71
E R2, symmetric 7 2
1, £ < 6-2, £ ~ r-4, then from (4.1), Condition F, and (4.17) (with £
replaced by 6-2-£) we obtain the estimate (4.38) which is (4.18).
78
Hence Theorem 3.5 (d) implies that
~ = Z(a2~), 70 = 8; S 2(a2~)(Y)dY,R
71 = 8; SR2 y(a2~)(Y)dy, 72 = 8; SR2
Thus, using (4.18), we have
T 2yy (a ~)(y)dy.
(4.67) 17s 1 ~ 8; J2IYls(1+IYI)-4-£lIa2~1I_4_iYR
~ M(£)lIa2~1I_4_£ ~ M(a,o,r,£,A,A)lI f ll_r a (s = 0, 1, 2).,Choose £ = min {(O-2)/2, r-4}; then this bound is independent of £ and
becomes (4.16). Again, when also 0 < £ < 1 and Ixl ~ 1, the
inequality (4.19) is a consequence of Lemma 3.1 (b) and (4.18).
Therefore, it is sufficient to prove that (4.17) holds for all
£ > o. Without lose of generality, we may assume, analogously to
(4.41),
(4.68) o < £ < min {1, (6-2)/2}, £ ~ r - 4.
The inequalities (4.24) still hold, and if ~ satisfies (4.66), then
~ E B 2 for all £ > O. Thus we may apply Theorem 2.4 (with u = 2+£)£+
and obtain
79
(4.69) 1Iif>II E+2 , 1 + 11Dc/>IIE+1, 1 + IIn 2if>IIE,1 + IIn3if>II
E_1, 1
s M(a:,<5,7",E,>.,AHIlif>1IE+2
+ IIn4if>IIE_2 a:,
+ IIfll }.-7",a:
It is, therefore, sufficient to derive instead of (4.17) the estimate
(4.70) 1Iif>IL 2 :s; M(a:,O,7",E,>',A)llfll .<;+ -7",a:
Suppose that (4.70) is false. Then there exist sequences {a }sm '
{bsm}, {csm}, {fm}, {if>m}, b om}' b 1m}, b 2m}, s = 1,2; such that all
a , b , c satisfy Conditon F (hence each c =0 in this case),sm sm sm sm
each f is in B , each if> is in B and a solution of (4.27) (withm -7", a: m c+2
c =0), with (4.66) holding for all {if> , 10
' 1 ,12m} and withsm m m nn
(4.71) 1Iif>lI c+2 = 1, IlfmIL7",a: :s; +.Therefore, by substituting (4.71) into (4.69) (applied to all pairs
{A f}) we have'I'm' m '
(4.72) IIif>m ll E+2 , 1 + IIDc/>mIIE+1, 1 + IIn2if>mllc, 1 + IIn3if>mIl E_1, 1 + IIn4if>mIlE_2,a:
:s; M(a:,O,7",E,>',A)(2).
Since now <5 > <5 - E > E + 2 (see (4.68)) we may pass to subsequences
to ensure that (4.46) holds. Then if> is a C4 solution of the equation
(4 .31) (with c = c =0 e =0)' and1 2' ,
2(4.73) ~(x)-~if>(x) = a1(x)en
if>(x)+b1(x)oDc/>(x)-aif>(x)
~ 0 as x ~ 00.
Since ~2if> = - (a-J)on4if> - bon3if> - con2¢ - doDc/> E BE
_O_2
C B_4
_E
, by
Theorem 3.5 (d) and Lemma 3.3, we have
and then, by [2, Lemma 3.2], there is a real constant 1,
1 1 J (~2if»(x)dx, such that ~¢(x) - 1loglxl ~ 0 as x ~ 00.= 2'1l" IR2
Hence, from (4.73), we have
80
~(x) - ~loglxl ~ 0 as x ~ 00 for some ~ E ~.
But since ~ solves the second order elliptic equation,
zazeD ~ + bz.Df =0,
the maximum and minimum of ~ on any circle about the origin must occur
on the boundary of this circle. We conclude that we must have ~ = 0,
~ == O.
Now we need to prove that ~ == O. We again have (4.33), (4.34);
and from (4.34), (4.68), Condition F (with (4.3)'), (4.72), (4.71), we
also have, analogously to (4.35),
(4.74) Igm" _4 - E,a s IIam-JILz_ZE,aIID4~mIlE_z ,a + IIbmILs_z€ ,aIIDs~mIlE_1 ,a
+ II cm"-4_u ,aIlD2~mll€ ,a + IIdmll_5_u ,all~ 1I€+1,a
+ lI em"-a_z€ ,all~mIIE+z, a + II f mIL4 _E
::;; M(A){M(a,6,T,E,A,A)e2} + 1.
Thus gm E B_4
_E
,a' and Theorem 3.5 (d) implies
~m = Zgm' ~Om = 8; J z gm (y )dy ,~
~1m = 8; J z ygm(y)dy, ~zm = 8; J z yyTgm(y)dy.~ ~
By making estimates as in (4.67), with (4.74) being applied, we find
that the sequences {~om}' {~1m}' {~2m} are bounded sequences of real
numbers, vectors, symmetric matrices, respectively. On passing again
to subsequences, we may assume ~om ~ ~O' ~1m ~ ~1' ~2m ~ ~2 for2 • 2x2some ~o E ~, ~1 E ~ , symmetrIc ~z E ~ . From Lemma 3.1 (b) we
have, for Ixl ~ 1,
loglxl
(4.75) I~m(x) - ~omlxI2(loglx!-1) + (210glxl-1)~lm·x2
[
loglxl - ~ +~2 Ixl2
- ~2m·x
lx2
Ixl2
:;;; M(E)l\g 1\ (1+lx\)-E l og(2+lxl).m -4-E
81
By substituting (4.74) and letting m~ 00, we see that (4.66) holds
as x ~ 00.
2Since ~(x) equals zero for all x E ~ , we have
(4.76)
Again, since ~ E B 2 is a solution of (4.76), the same modifiedE+
theorem of [2] implies that
(4.77)
for some ~ E ~ and some harmonic polynomial
(4.78)
Then (4.66) and (4.77) imply that (4.66) holds with ~(x) replaced by
pz(x) + ~loglxl· Since the growth rates of x~, xlx2' x~, xl' x2 '
loglxl, IxI 20oglxl-1), (21cglxl-1)xl and (2Ioglxl-1)X2
are variant as
x ~ 00 from different directions, it is easy to show, from (4.66),
(4.77) and (4.78), that
(4.79) rll
= r1 2 = r 2 2 = ql = q2 = 10 = 0, ~l = 0, 1 = ~11 + ~22'
and
2 2
(4.80) (XII) ( x2 1) - 2~
xlx 2~ 0,P - ~ - 2"111 - ~ - 2"12 2 12
Ixl2
as x ~ (Xl, where [1'11 1'12] := 1'2'1'12 1'22
Now by letting x = (xl' O)T~ (Xl and
have
rX = (X
2'0) ~ (Xl in (4.80), we
82
1P ± 2(1'11 - 1'12) = 0,
and then combining with l' = 1'11 + 1'22' we get
(4.81) 1'11 = 1'22 = 1'/2, P = O.
Thus, from (4.80) (with (4.81) applied), we also have 1'12= O.
Hence, from (4.78), (4.79), (4.81), we have P2 =0; and then,
from (4.77), we see that ~(x) - 1'loglxl vanishes at infinity. But
since ~ solves the second order elliptic equation (4.76), the maximum
and minimum of ~ on any circle about the origin must occur on the
boundary of this circle.
4> =O.
We conclude that ~ = 0 (hence ~ - ~ - 0), '11- '22- ,
Therefore, (4.46) yields DS~ ~ 0 in B~ ,s = 0 1 2 3'f'm v-E-S ""
4. Again from (4.34) and (4.71) we have estimate (4.50). Hence
which implies
as x ~ 00. Finally, from Lemma 3.1 (a),
14> (x l ] :;; M(E)llg II O+lxl)2 I og(2+lxl),m m -4-£
that 114> II 2 ~ 0 as x ~ eo, contradictingm E+(4.71) . I
Theorem 4.1 can be restated as the following uniform way:
Theorem 4.1' . Suppose Condition F holds, that fEB for some T,-T,O:
T > 4, and that 0: is tha same as in Condition F. Assume that ~ is an
4-nentire C4 solution of e4.1) in ~n en~2) such that 4>ex) - L l' .Dsrex)ss=o
vanishes at infinity for some constants l' of the same dimensions ass
-- ---------- --------
83
h I s M(a,6,1",A,A)llfll ,s -1",a
and for any £ > 0 (£ may equal zero whenever n is odd or n ~ 4+1),
4-1
s~o"osq,l!max(0, 4-n)+£-s, 1 + lIo4q,lImax( 0, 4-n)+£-4,a
~ M(a,6,1",£,A,A)l!fl! .-1",a
If, moreover, £ < 8 - max{O, 4-n}, £ ~ 1" - 4, then
I!Li4 / 2q,IL4_£ ,a ~ M(a,6,1",£,A,A)lI fll_1",a;
and if also 0 < £ < 1 and Ixl ~ 1, then
if n is odd or n ~ 4+1,
if n is even and n ~ 4.
5. EXISTENCE OF SOLUTIONS
We mow demonstrate the existence of entire solutions of the
nonhomogenous equation,
(5.1) L<P:= ao0 4q, + bo03q, + co02q, + doD</> + eq,
= L2L1q,
:= (a200
2+ b
200 + c
2)(a1002q, + b
1°oq, + cIq,) = f,
with certain prescribed behaviour at infinity. Again we assume
Condition F on the cofficients of L.
THEOREM 5.1. Suppose Condition F holds on the cofficients of Land
that fEB for some r > 4.-1",a
(a) For n ~ 5, there exists a unique entire C4 solution q, of
(5.1) such that q, vanishes at infinity. Moreover, for this solution q,
the bound (4.5)-(4.7) of Theorem 4.1 (a) hold.
84
(b) For n = 4, there exists a unique entire C4 solution ~ of
(5.1) such that ~(x) - ~ologlxl vanishes at infinity for some constant
~o. Moreover, for this solution ~ the bounds (4.8)-(4.11) of Theorem
4.1 (b) hold.
(c) For n = 3, there exists a unique entire C4 solution ~ of
(5.1) such that ~(x) - 70lxl + 71-TiT vanishes at infinity for some
constant ~o and constant vector 71
, Moreover, for this solution ~ the
bounds (4.12)-(4.15) of Theorem 4.1 (c) hold.
(d) For n = 2, there exists a unique entire C4 solution ~ of
(5.1) such that
(5.2)
loglxl
~ 0,
as x ~ 00, for some constant 70
, constant vector 71
and constant
symmetric matrix 72
, Moreover, for this solution ~ the bounds
(4.16)-(4.19) of Theorem 4.1 (d) hold.
Proof: In all cases (a)-(d), the uniqueness of the solution and
bounds (4.5)-(4.17) are consequences of Theorem 4.1. Therefore, we
only need to establish existence of a solution.
(a) Let E a be fixed number such that
o < E < 1, E ~ min {8/2, T-4}.
We consider a Banach space ~ defined by
~ := {~ E C4(~n) : ~2~ E B , ¢(x) vanishes at infinity}-4-E,o:
with norm
(5.3) 1Iif>1I* := 11112if> 1I_4_ E 0:.,By Theorem 4.1 (a), applied to L =112, for any if> in B we have
(5.4) 11if>llo,1 + 1ID4>1I_1,1 + IID2if>1I_2,1 + IID3if>1I_3,1 + IID4if>11_4,0:
::; M(n,0:,E)1I112if>1I_4_E 0:',
and for Ixl ;a: 1,
(5.5) Iif>(x) I ::; M(n,0:,dIl112if>1I_4_E o:(l+l x l )-E.,
From (5.4)-(5.5), it is easy to verify that B is indeed a Banach
space (see the similar proof in (b)).
We also consider two families of linear operators defined by
85
(5.6)
and
(5.7)
Setting
we have
and
2(5.9) Itif> = L2(a1t·D if> + b1t·D¢ + c1t¢)·
=: (at-J).D4if> + bt.D3if>
+ Ct · D2¢ + dt°D¢ +
We first note that, for 0 ::; t ::; 1,
(5.10) IlascIII_6,0: = tllas-III_6,0: ::; A,
IIDsa
l t 11-s-6 a' IIDsb
l t"-I-s-o Ct', ,
and then, from (5.5), Lemma 2.3, (4.3)'
Ilbstll_I_O,0:' IlcstlL2_0 ,0: s A,
IIDSCltll_2_s_o Ct::; A (s = 1,2),,
and (5.4), if if> E B,
(5.11) III t4>II_4_€,a:;;; MII4>lI o,a + IID4>II_ I,a + IID24>IL2,a
+ IID34>IL3,a + IID44>1I_
4,a] + 1I~24>IL4_E,a
:;;; M(n,a,€,A)II4>II*·
Similarly, if 4> E S,
(5.12) IIPt4>II_4_€ a::; M(n,a,E,A)II4>II*·,Hence, both :Pt and It, 0 :;;; t :;;; 1, map S into B We also get
-4-€,a
from Condition F that
(5.13) Cst = tc s ::; 0 if n ~ 3, and Cst =0 if n = 2 (s = 1, 2);
and, for any x, e ERn,
86
(5.14) ast(x)e.e = (l-t)leI2
+ tas(x)e·e
~ (l-t)leI2
+ tAlel2~ min{I,A} lel 2 (s = 1, 2).
Therefore, Theorem 4.1 can be applied, and for all 4> E Sand 0 ::; t ::; 1
we have estimates
(5.15)
and
(5.16) 114>11* s M(n,a,6,'T,).,A)IIIt4>11 .-'T,a
According to Theorem 5.2 in [11], the Schauder continuation method
applies. Theorem 3.5 (a) and Theorem 4.1 (a) assert that the operator
2:Po = ~ maps S onto B . Hence, from (5.15), any :Pt, 0 ::; t ~ 1,-4-E,a
does the same and, in particular, :PI = L26
= I o maps S onto B .-4-E,a
Then, from (5.16), applying the Schauder continuation method again, we
conclude that II = L2LI = L maps S onto B_4
_E
,a '
B , the proof of (a) is complete.-4-€,a
(b) Let € be chosen and fixed so that
Since B C-'T,a
o < € < 1, E:;;; min {6/2, 'T-4}.
87
We define a now as the Banach space
(5.17)
at infinity for some real constant ~o}'
with the norm (5.3). By Theorem 4.1 (b), applied to L = ~2, for any ~
in a with corresponding ~o we have
(5 .18) I~o I :;:; M(a, e ) 1I~2~1I_4_E a',(5.19) 1I~IIE,l + 1ID4>II E_1,1 + IID2~lIc_2,1 + IID3~IIE_3,1 + IID4~IIE_4,a
s M(a, E) 11~2~IL4_c a',
and for Ixl ~ 1,
(5.20) I~(x) - 'Yologlxll :;:; M(a,E)II~2~11_4_E a(l+lxl )-E l og(2+lxl).,(These results also follow from Theorem 3.5 and Lemma 3.1.) In order
to verify that a is a Banach space, we consider a sequence {~ },m
Cauchy in the norm (5.3) of a, with corresponding associated constants
{~om}· Inequalities (5.18)-(5.20) show that {~m} is also Cauchy in
the norm determined by the left hand side of (5.19), that the sequence
t~om} is a Cauchy sequence of real numbers, and that the limits ~ and
~ of {~ } and {~ } satisfy an inequality similar to (5.20), therebyo m om
implying that ~(x) - ~ologlxl vanishes at infinity. Hence ~ E a, and
we conclude that a is indeed a Banach space with respect to the norm
(5.3).
We again consider the families of operators Pt and It, 0 ~ t ~ 1,
defined by (5.6)-(5.9), with (5.10), (5.13), (5.14) also being valid.
Then, from (5.10), Lemma 2.3 and (5.19), we have, analogously to
(5.11)-(5.12),
88
(5.21) IIPt4>II_4_£,a' III t4>IL4_£,a:;; MII4>II£,a + 11D4>1I£_l,a + IID24>1I £_2,a
+ IID34>1I £_3 a + IID44>1I£_4 a] + II Ll24>IL4_£ a, , ,
:;; M(a, e ,A) IILl24>1I _4_£ a =M(a, e ,A) 114>11*.,This shows that both Pt and It, 0 :;; t :;; 1, map a into B . Then
-4-£,a
Theorem 4.1 (b) (with T = 4 + £) gives the bound, for all 4> E a,
(5.22) 114>11* :;; M(a,6,£,>',A)emin{IIPt4>II_4_£ a' II I t4>1L4_£ a}', ,Theorem 3.4 (b) and Theorem 4.1 (b) assert that Po = Ll2 maps a onto
B .-4-£,a
= L. Since B C B 4 ' the proof of (b) is complete.-T,a - -£,a
(c) Let £ be a fixed number such that
o < e < 1,
Define a as the Banach space
£:;; min {(O-1)/2, T-4}.
a := {4> E c4(~3) : Ll2
4> E B_4_£,a' 4>(x) - ~olxl + ~leTiT vanishes
at infinity for some constant ~o and constant vector ~1}
with the norm (5.3). The rest of the proof is similar to the proofs
that we presented in (a) and (b).
(d) Let £ be chosen and fixed so that
o < £ < 1, £ ~ min {(O-2)/2, T-4}.
Define a now as the Banach space
a := {4> E c4(~2) : Ll24> E B 4 and (5.2) holds as x ~ 00- -£,a
2 . 2X2}for some ~o E ~, ~1 E ~ , symmetrIC ~2 E ~
with the norm (5.3). Again, the rest of the proof can be obtained by
following the procedure of (a) or (b). I
In order to discuss the existence of entire solutions of (5.1)
with polynomial growth at infinity, we require the following
---- --------------
modification of Condition F.
89
Condition MF The coefficients of L satisfy Condition F, and moreover,
there exists a nonnegative integer m with m' := m - max{O, 4-n} ,
such that
From (5.1) and Lemma 2.3, we see that (5.23) implies the
following condition on the coefficients a, b, c, d and e:
(5.23)' lIa - JILm, -6,0:' IIbll_m, -1-6 ,0:'
IIdll_m' _3_6 0:',
IIcll_m' _2_0 0:',
lIe!!_m' -4-6 0: s M(A).,
After replacing Condition F by the stronger Condition MF, we will
have the following extension of Theorem 5.1.
THEOREM 5.2. Suppose Condition MF holds on the coefficients of L,
and that fEB for some 7 > 4. Then:-7,0:
(i) For any biharmonic polynomial p, with (degree p) ::; m, there
exists a unique entire C4 solution ~ of (5.1) such that
(a) if n ~ 5, ~(x) - p(x) ~ 0 as x ~ 00;
(b) if n =4, ~(x) - p(x) - ~ologlxl ~ 0 as x ~ 00 for some
constant ~o E JR;
(c) if n =3, ~(x) - p(x) - ~ologlxl + ~1.TiT ~ 0 as x ~ 003
for some ~o E JR and ~l E JR ;
(d) if n = 2, ~(x) - p(x) - 10IxI2(loglxl-l) + (2Ioglxl-l)11ex
21 Xl X1X2
[
l og IX I - '2 +~ I 2 ]
- 'Y,. X,X. Log lx ] :11. + X: ----+ 0
Ixl 2 2 Ixl2
as X~ 00 for some 10 E~, 11 E ~2, symmetric '2 E ~2X2.
(ii) If ~ is an entire C4 solution of (5.1), with ~ E Bandm+O'
o ~ 0' < min {I, o-max(O,4-n)}, then there exists a unique
90
biharmonic polynomial p, with (degree p) ~ m, such that (a)-(d)
above hold.
Proof: (i) The proof of the uniqueness of ~ again follows from the a
priori bounds of Thorem 4.1. By Lemma 2.3 (e), we have p E B ,Dp Em,l
B 1 l' D2p
E B 2 l' D3p
E B 3 l' D4p
E B 4 l' Writingm- , m- , m- , m- ,
Lp = A2p + (L - A2)p = = (a-J)eD4p + beDsp + ceD2p + deDp + ep,
and applying Condition MF (with (5.23)' and (4.3)' being applied), we
have, for 0 < E < 0 - max{O, 4-n} ,
(5.24) IILpll_4_E,a:5 lIa-JILm_E,aIID4pllm_4,a + IIbll_m_l_E,aIlDspllm_s,a
+ II cILm_ 2_E allDzpllm_z a + IIdll_m_ 3 _ E a11DPllm_1 a, , "+ Ilell_m_4_E a11pllm a', ,
s M(A) [IID4pllm_4
1 + IIDspllm_s 1 + IIDzpllm_2 1, , ,
+ IIDPllm_1 1 + IIp11m 1]', ,and hence Lp E B . By Theorem 5.1, there exists a unique entire
-4-E,a
C4 solution ~ of the equation
L~ = f - Lp
such that if n ~ 5, ~(x) ~ 0 as x ~ 00; if n = 4, ~(x) - 10loglxl ~
91
x+ "( e'"T::T ~ 0 as1 'XI
o as x ~ 00 for some 10 E~; if n = 3, ~Cx) - 10lxI
x ~ ~ for some 10 E ~ and 1 1 E ~3; and if n = 2,
C5.25) ~Cx) - 10 Ix I2Cloglxl-1) + C2loglxl-1)11ex
2
[
loglxl - i + 1:~2- 1 2 -
x1x2
Ixl2
X1X2
log IXI
I
:
I:
+ -~J ---. 02 IxI2
as x ~ 00 for some 10 E~, 11 E ~2 and symmetric 12 E ~2X2. Setting
~ := ~ + p, we obtain the desired solution ~.
Cii) The proof of the uniqueness of p follows from the a priori
bounds of Theorem 3.5 applied to a2.
In order to show existence of p, we observe that if ~ is a C4
solution of C5.1), with ~ E B ,then Theorem 2.4 implies thatm+u
~ E Bm+u , I ' ~ E Bm+u_1 , 1 ' D2~ E Bm+u_2 , 1 ' D3~ E Bm+u_3 , 1 ' and
D4~ E B 4. We also havem+u- ,a
a2~ ; ~ _ CL-a2)~ = f - Ca-J)eD4~ - b.D3~ - ceD2~ - de~ - e~.
Choose € so that 0 < € < r-4 and € ~ min {1, o-maxCO,4-n)} - u. Then,
from Lemma 2.3 and Condition MF,
lIa2~1I_4_f,a ~ II fll_ 4_f,a + lIa-JII_m_f_u,aIlD4~lIm+u_4,a
+ Ilbll_m_l_€_u,aIlD3~lIm+0"_3,0: + IIcll_m_2_f_0",o:iID2~lIm+0"_2,0:
+ Ildll_m_ 3_€_u,0:11D4>lIm+0"_1 ,0: + Ilell-m-4-€-u,all~lIm+u,a4
s Ilfll_r 0: + MCA) L IIDs~llm+O"_s a', s=o '
which implies a2~ E B 4 . Therefore, by Lemma 3.3 the potential- -€,a
Zca2~) is C4 in ~n, with a 2CZCa2¢)) = a 2¢. Moreover, Ca)-Cd) of Lemma
3.3 are satisfied by Z(~2~). The function
92
(5.26)
is biharmonic in ~n. If n ~ 4, then, since ~ E B ,Z(~2~) E B ,m+O' 0'
Lemma 3.4 asserts that p is a polynomial with (degree p) s m. The
above reasoning applies for the cases of n = 3, m ~ 1 and n = 2, m ~ 2
since Z(~2~) is in Bl and B2
+0' , respectively. For n = 3, m = 0, we
have ~ E BO', with Z(~2~) - 10lxI + 7l .TiT vanishing at infinity, and
then Lemma 3.4 implies p(x) = q.x + Po' a polynomial with degree S 1.
But then
~(x) - q.x - Po - 10l x l + 1 l .TiT ~ ° as x ~ ~,
with ~ E BO', 0' < 1. It follows that 10 = 0, q = 0, and p(x) = Po is
a polynomial of degree zero. For n = 2, m = ° or 1, we have ~ E Bm+O'
C Bl and (5.25) holds with ~ replaced by Z(~2~). Then Lemma 3.4+0'
implies that
2 2p(x) = rllx l + 2r 12xlx 2 + r 22x2 + qlx l + q2x2 + Po'
a polynomial of degree S 2. Hence, from (5.26), (5.25) for Z(~2~),
~(x) - rllx~ - 2r 12xlx2 - r22x~ - qlx l- q2x2 -Po - 70Ix I2Cl og lx l- 1)
2
[
loglxl - ~ +~2 Ixl2
+ (2Ioglxl-1)1l•x - 12• xlx2
Ixl 2
~ ° as x ~ 00.
If m = 1, then ~ E Bl+O', and hence 70 = r l l = r 12 = r 22 = 0, and
p(x) = qlx l + q2x2 + Po is a polynomial of degree S 1; if m = 0, then
~ E BO', and it follows that 11 = 0, ql = q2 = 0, and hence p(x) =Po'
(5.27)
a polynomial of degree zero. Finally, writing ~ - p =Z(~2~) gives
(a)-(d) . I
Remark. Theorem 5.1 and 5.2 can also be restated in a uniform way as
we did for Theorem 4.1. For example, the statements (a)-(d) of
Theorem 5.2 can be writen in the following uniform way:
4-n~(x) - p(x) - L , eDsr(x) ~ 0 as x ~ 00
s=o s
for some constant ~ of the same dimensions as DSr.s
For the special case of the homogeneous equation,
93
(5.28)
Theorem 5.2 implies the following:
COROLLARY 5.3. Suppose Condition ~w holds on the coefficients of L.
Then the space of entire C4 solutions of (5.28) which are in B formsm
a finite dimensional vector space, of dimension the same as that of
the space of biharmonic polynomials of degree no greater tilan m.
There is a one-to-one correspondence between the members of these two
spaces, determined by (5.27) or, equivalently, by statements (a)-(d)
of Theorem 5.2.
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