Math. Meth. in the Appl. Sci. 4 (1982) 131 - 142 A M S subject classification: 35 P30.35 ROS, 35 R35

A Discontinuous Nonlinear Eigenvalue/Free Boundary Problem

R. Alexander, Troy

Communicated by P. H . Rabinowitz

We study the problem

- A u = dH(u - 1 ) in R

u = g in B Q .

(His the Heaviside unit step function) in spherical domains R of arbitrary dimension. When g = 0, there are two branches of radial solutions; for small nonzero g there are solutions near the corresponding radial solution. Moreover, the set where u = 1 is in all cases an analytic hypersurface.

0 Introduction

Let SZ be the unit ball in FIN, N > 1, and S its boundary, the unit sphere. We want to solve the problem

(0.1) - o u = AH(u - 1) in a

u = g on S ,

where H denotes the Heaviside unit step function and g is a given function. The presence of the Heaviside function in (0.1) requires that the concept

of a "solution" of (0.1) accommodate jumps in the second derivatives. We shall find solutions u E C'*"(fi) with the property that the set (x E SZ: u ( x ) = 1 } is a smooth hypersurface on each side of which the equation A u + I = 0 (on the side where u > 1) or A u = 0 (on the side where u < 1) is satisfied in the classical sense. This hypersurface is called a free boundary because it is not given beforehand but is determined by the solution u. Since u E C'*", u and its gradient will be continuous across the free boundary.

We refer to the reduced problem (0.1) when g e 0, to the perturbed problem when g does not vanish identically. Our approach is first to obtain all solutions of the reduced problem, and then by local methods to solve the perturbed problem. Thus the condition that g be sufficiently small will always be in force.

0170-4214/82/01 0131 - 12 $02.40/0 0 1982 B. G.Teubner Stuttgart

132 R. Alexander

This problem has been studied by Fleishman and Mahar [4] in the case N = 2 for a continuous g satisfying 0 Q g Q & for a constant &sufficiently small. By a monotone iteration method they obtained solutions of (0.1) near the upper solutions (see Section 1) of the reduced problem. In the present work, by applying a method introduced in [l], we shall eliminate the restriction that g be non negative, we can perturb both upper and lower solutions of the reduced problem, and we shall find no difficulty in working in any number of dimen- sions. In addition we demonstrate the regularity of the free boundary.

The method described here and in [l] may be applied to other free boundary problems. That (0.1) is not merely a toy problem is shown by the fact that the simple transformation converts it into a version of a model studied in the theory of plasma confinement. If u is a solution of (0.1) with g a constant, and if I > 0 is a given constant, let

y = I / j d o

u = y(u - 1) S

I = - y A ;

then u is a solution of

A u = I H ( - v ) i n 8

u = y(g - 1) o n S 5 au/an d o = I , S

which is the problem studied in [12] with u- replaced by H ( - u ) in the differen- tial equation. Other right-hand sides proposed in models of plasma confinement are given in [2]. Further work on differential equations with discontinuous non- linearities may be found in [2], [3], [ill, and references cited in those works.

In Section 1 of this paper we derive the solutions of the reduced problem, which are all radially symmetric. In Section 2 we show how the perturbed problem is equivalent to a nonlinear integral equation for the free boundary. Section 3 solves that integral equation, and Section 4 concludes with the proof that the free boundary is analytic.

I am grateful to Bernard Fleishman for a number of stimulating dis- cussions, and to Bruce Turkington for pointing out the transformation given above.

1 Radial solutions

This section gives the radial solutions of

- AU = AH(u - 1) in 8

u = o on S . (1 -1)

A Discontinuous Nonlinear EigenvalueIFree Boundary Problem 133

By a solution of (1.1) we understand a function u E C'(fi) n C2 (Q\{x: u ( x ) = 1)). Of course u = 0 is always a solution. By Hopf's maximum principle, a positive radial solution u = u ( r ) can take on the value 1 at only one value of r, say r,, and must have adar < 0 there. Hence we obtain all radial solutions of (1.1) by finding u, A and ro so that the differential equation

O < r < r , r , < r < l

(1.2) -rl-Na/ar(rN-l au/ar) =

the boundary conditions

(1.3) au/ar(o) = o ~ ( 1 ) = o

(1.4) U(T , - 0) = u(r0 + 0) = 1

and the transmission conditions at the interface

au/ar(r, - 0) = au/ar(r, + 0) are all satisfied.

The results for N = 2 in the remainder of this section are given in [41. In order for all of (1.2), (1.3), and (1.4) to be satisfied, a compatibility

- (I/2) to' log r, = 1,

condition relating I and ro must hold, which is found to be

(1'5) r t - ri + N ( N - 2 ) / I = 0 ,

As an equation in r,, (1.5) has either one or two roots between 0 and 1, when I is respectively equal to or greater than A*:

N = 2 ,

N > 2 .

I* = 4e, N = 2 ,

N ( N - 2) A* = . N > 2 . (1.6)

Finally, when A 2 I* and r, is chosen to satisfy (1 S), the solution u is given by

ro' - 72) - (A/2) ro' logro, - (A/2)ro'logr,

0 < r < ro r, Q r < 1

N = 2 ,

For A > A*, call r; and r i the roots of (1 3, with r; > r;. Then u given by (1.7) with ro = r; is pointwise greater than u with ro = r;; we call the former an upper, the latter a lower solution.

We remark that since H ( u - 1) is monotone and not decreasing, the result of [6] shows that all positive solutions of (1.1) are radial.

134 R.Alexanda

2 An equivalent integral equation

The natural way to solve (0.1) seems to be to use the Green’s function to convert it into a nonlinear Hammerstein integral equation,

(2.1) and indeed global solution branches have been obtained for similar equations with discontinuities, see [9], [2].

Our goal is different. Having a solution of the reduced problem, we study the effect on the solution of perturbations in the boundary values, and indeed will be able to calculate that effect to first order. But the right side of (2.1) is a discontinuous operator, and seems to be not amenable to a rigorous perturbative approach, although a formal scheme was devised in IS]. Instead we change our point of view, as in [l], and derive an integral equation to be satisfied by the free boundary.

Henceforward we use spherical polar coordinates (r, o), with o standing for the angle variables.

The solutions of the reduced problem (1 . l) have a free boundary which is a sphere given by

u = AGH(tc - 1) ,

u(r0, 0) = 1 . If a perturbation theorem is true, then for g sufficiently small a solution of (0.1) should have a free boundary given by

(2.2) u(rg + b(o), w) = 1

for some function b (w), depending on g and A , which we call the perturbation in the free boundary.

Now, a solution of (0.1) with free boundary given by (2.2) solves the linear Poisson problem

At this point we specify the class of boundary functions g which shall be admitted. Since the right hand side of the differential equation (2.3) is in every Lp, we can get u E W2*p forp > N, hence u ~C’*”(f i ) , provided g is the trace of a W2*P function, that is g E W2-’/PiP (S) . If we choose a less regular class of bound- ary values, it remains true that u E W;:(Q), and the results to follow continue to hold; for simplicity we fa a p > Nand carry out the proof for g E W2-’/PJ’.

Let then A > A*, let ro satisfy (1.5), let g E W2-”p*p(S) be given with maxIg(w)l< l,andassumethatbEC(S)satisfiesO < r o + b(o) < l , o ~ S . For this choice of A, ro, g and b let u be the solution of (2.3), and assume that (2.2) holds. Then by the maximum principle u (r, o) > 1 for r < ro + b (o) and u (r, o) < 1 for r > ro + b(o) (cf. [l], Proposition 1) and it follows that u is a solution of (0.1). The condition that the solution of (2.3) satisfy (2.2) gives us the integral equation for 6, which we write down presently.

A Discontinuous Nonlinear EigenvaludFree Boundary Problem 135

First, however, note that the radial solutions (1.7) satisfy Ou/Or(ro) = - Aro/Nc 0; if g is small enough, it follows that the solution of (2.3), being close in Wztp, hence in CIS', to a radial solution, will also have 0u/ar (ro + b (w), w) bounded away from zero. This implies by the classical implicit function theorem that b e C'*"(S).

We now give the integral equation for b determining solutions of (0.1). With I , ro, g as above, choose b E C ( S ) , 0 < ro + b(w) < 1, and let u be the solution of (2.3). Then

(2.4)

defines a mapping of a neighborhood of (0, A , 0) E W2-1/p*p(S) x R x C(S) into C(S) .

Writing P for the Poisson kernel, G for the Green's function, and x b for the characteristic function on the right side of (2.3) we have

F(g,A,b)(w) = u(ro + b(w) , w) - 1 w E S

(2.5) u = Pg - A G x b ,

or

F(g, A, b ) (0)

(2.6) = P(ro + b(w), w, w') g(w') d o ' S

ro + b(w') - A j d o ' ( r ' )N-ldr 'G(ro + b ( o ) , o , r : o ' ) - 1 .

S 0

3 Solution of the integral equation

We begin directly by stating the main theorem.

Theorem 1 Let 1 > I* , and let ro have one of the two values satisfying (1.5). There is a neighborhood U of ( 0 , i ) in Wz-' /p*p(S) x R and a unique continuously differentiable mapping B : U -, C ( S ) such that B (0, I ) = 0 and F(g, A, B (9, A ) ) = ofor (9, A ) E U.

From this, based on the discussion of Section 2, we immediately deduce our result on solutions of (0.1):

Theorem 2 Let 1, robe as above, and let Ube the neighborhood given by Theorem 1. Then for (9, A ) E U, (0.1) has a solution u depending continuously on (g , A ) , given by

u = pg - A G X B ( g . L ) *

Theorem 1 follows from two lemmas and the implicit function theorem.

Lemma 1 For each 1 > A*,F b a continuously differentiable mapping of a neighborhood of (O,I, 0) in Wz-'/P,P(S) x R x C(S) into C(S).

136 R. Alexande~

Lemma 2 For any A in the neighborhood of ( O , x , O ) given by Lemma 1, D3F(0, A , 0) E L (C(S) ) has u bounded inverse.

P r o o f o f Lemma 1 .Letx > A*. WeshowthatthepartialderivativesDjF,j = 1, 2, 3 exist and are continuous in a neighborhood of ( O , x , 0). For j = 1 this is obvious because Fdepends affmely on g . The proof f o r j = 2 is elementary and is left to the reader. As usual, it is clear what D3F(g, A, b) ought to be; the work is all in proving that the linearization of (2.6) about a given b is a Frechet derivative. We assert, that D, F is given by

a p ar

= b d w ' - (ro + b ( o ) , o , 0') g ( o ' )

-Ajdo ' ( i r , + b(o')]"-'G(r0 + b ( o ) , o , r o + b(o ' ) , o ' )B (o ' ) ) .

Notice that the bracketed expression in the second line of (3.1) is just 8u/8r(ro + b ( o ) , a), where u is the solution of (2.3). Once (3.1) is verified it follows easily, as in [l], that (g ,A, b ) + D3F(g, A, b) is a continuous mapping of a neighborhood of (0, A , 0) into L (C(S)), and that the integral operator in the third line of (3.1) is compact. To prove (3.1), imitate the proof of the Leibnitz rule for differentiation of integrals depending on a parameter. Denote by L the right side of (3.1). Then

S

.

F(g,A,b + B ) ( o ) - F(g,A,b)(w) - LB(0)

a + b@') + A j d w ' j ( r ' )N- 'd r 'G(rO + b ( o ) , o , r : o ' ) S 0

A Discontinuous Nonlinear EigenvaludFree Boundary Problem 137

The three terms in the second line are o ( ll/?llm) as 11j?11.. -+ 0 by Taylor’s theorem and uniform continuity. When the following term is added and subtracted in the third line, -

a + NU‘) j A j d o ‘ ( r r ) N - l d r ‘ G(ro + b(o) + b(o), o, r’, 0’)

S 0

- the thud through sixth lines of (3.2) become

-AGXb(rO + b(w) + B ( w ) , o ) + A G X b ( r 0 + b(a) ,W)

+ AG,x*(ro + b ( o ) , w ) B ( o )

S a + Nw’)

- 1 { d o ’ ([ro + b(o‘)lN-’ G(r0 + b ( ~ ) , ~ , r o + ~ ( u ’ ) , w ’ ) B ( w ‘ ) ) S

The first line of (3.3) is again o(llBllm) for IIB\lm -+ 0 by Taylor’s theorem and uniform continuity of G,yb and its first derivatives. In the remaining integrals we add and substract another term, leaving

a+ b ( w ’ ) + b W ) - A J d o ’ j ( r ‘ ) N - l d r ’

s a + b b ’ )

* [G( ro + b(w) + ~ ( W ) , U , ~ : W ’ ) - G ( r 0 + b(w) ,w , r ‘ ,o ’ ) l

) 1

a+b(w’)+b(w’) , - A j d o ’ [( j ( r ’ )N- ld r ’G( ro+ b(w) ,w , r ’ ,o ’ )

- [ro + b(o’ ) lN-’ G(ro +‘b(o),w,ro + b(o ’ ) ,o ’ )P(w‘ ) .

(3.4)

S a+ NU’)

We estimate only the second integral in (3.4); the first is similar. Note first that the second integral may be written

5 ( r r ) N - l d r ’ + b(o) ,w,r’ ,o‘) a + b(w‘) + b(w’ )

j d o ’ S a + b(w‘)

We show that as a function of w the integral (3.5) is o(llBIIm) for llBllm -, 0. We choose a real number y between 0 und 1 depending on 11/311, in a manner to be determined. For a fixed o E S let E denote the region of integration in (3.5), and D the b d of radius y about (ro + b(w) , a). Writing Q for the integrand we have

S Q = j Q + j Q . E EnD E n D C

138 R. Alexander

There are constants C' and C not depending on o such that

,.2-NrN-l d r < c y 2 N > 2 ,

E n D C' jlog(l/r) r d r < Cy210g(l/y) N = 2 . (3.6) I j Q l G r! Outside D, estimate the integrand by the mean value inequality to obtain, with new constants C' and C independent of O,

where we have used max r = y and E nDC

Now combine (3.6) and (3.7), giving

< c, Y 2 log(l/Y) + c2 Y1-NllPl15 IP 1 then choose y = IIPIlL-6 for a small constant 6 > 0, and it is easy to see that

P r o o f o f Lemma 2. From the expression (3.1) for D3F, observe that

Q = o ( I[ /311-) as + 0. This completes the proof of Lemma 1. IE 1

ww, B(O) = au/ar(ro, w ) P(W)

- A#- , j d o ' G(r0, O, t o , o ' ) ( w ' ) ,

where u, the solution of (2.3) with g = 0, is given by (1.7). Hence 8u/8r(ro, o) = -Aro/N, and

D,F(O,A,O) @(a) = -Are -1 + #-2K P ( w ) I: 1 where K is the integral operator in C ( S ) with kernel

As we remarked previously, K is compact, so D3F(0, A , 0) is invertible provided

(3.8) 1/N + r r - 2 Q # 0

for any an eigenvalue of K.

A Discontinuous Nonlinear Eigenvalue/Free Boundary Problem 139

When N = 2 we find 2 4 [I - cos (o - w’) ]

1 - 2r,Zcos(w - a’) + r: G(ro, 0, ro, a’) = ( 1 / 4 x ) log

21 1 - ro m

= ( 1 / 2 x ) logr, - C 1-1 1

so that eigenvalues and eigenfunctions are

a, = logro @ o = 1, 1 = 0

Now (3 .8) requires 1 /2 + at # 0, I = 0 , l , 2 , . . . . This condition is only violated when I = 0, a, = logto = - 1 / 2 , which implies 1 = 1* = 4e. This proves Lemma 2 when N = 2. In the case N > 2 we use the expansion of the Green’s function in spherical harmonics. Although this expansion for N = 3 is familiar, it may be less so if N > 3, and we give the elementary derivation in the Appendix. The result is

in which the reader who skipped the Appendix will want to know that the Ylm are orthononnd spherical harmonics of degree I, ON is the volume of the unit ball in N dimensions, and

( I + N - 3) ! n(N,I) = (21 + N - 2) I ! (N - Z)!

is the number of spherical harmonics of degree 1 in dimension N. The series converges uniformly in 10 - w’ 1 2 q for any q > 0. From (3.9) we read off the eigenvalues and eigenfunctions of r c - 2 K to be

It follows that (3.8) must be satisfied unless I = 0, for if I 2 1

= 0, implies ro = ( 2 / ~ ) ” ( ~ - ~ ) , or I 1 - On the other hand if 1 = 0, - - A = A*. This completes the proof of Lemma 2.

N N - 2

140 R. Alexander

4 Regularity of the free boundary

Let u be one of the solutions of (0.1) as given in the previous section. Let

r = {(r, w) : u(r, 0) = I} = {(ro + b(w), w) : w E S}

be the free boundary, which by the considerations of Section 2 we know to be a C'*'hypersurface. We show that Tis in fact analytic.

Consider a small ball B about a point xo = (ro + b(wo), wo) E rtrans- lating coordinates so that xo = 0 and using the rotational invariance of the Laplace operator, and writing r for r n B, v = u - 1, we have

A V + I = 0 in B+ = {X = (XI, ..., xN) E B : x N > 0)

AV = 0 in B- = {X = (x, , ..., XN) E B : x N < O}. a v a v

ax, a X N v E P(B+ u r u B-1, v = o on r, - # 0 on r, - (0) > 0

The value of A turns out to be irrelevant; it is the nonvanishing derivative of v in the direction normal to r which is decisive.

As in [8] we introduce the zeroth order hodograph transformation

yj = xj j < N

Y N = V ( X ) X € B .

If B is small enough to begin with, this is a 1 - 1 mapping of B onto some neighborhood U of the origin in y-space:

B+ -+ U+ = (.V E ~ : Y N > o} B- -+ u- = (.V€ u:J" < o} r +Z = C V E U : ~ ~ = 01.

Now define we) = xNfory E U; then !u E C'*a(U- u 2 u V) , and Vsatisfies a nonlinear equation in U+ :

(4.1) f ( w , D w , D 2 w ) + A = 0

where, writing subscripts for partial derivatives and summing j from 1 to N - 1,

Moreover, w satisfies f ( w , D p, D 2 w ) = 0 in U-. Writing y = Ql, . . . ,yN-l ,yN) = @:yN), we define for y E U+

@e) = V b : -yN) *

Then @ satisfies the equation in U': (4.2) f (@>D@DD2@) = 0

A Discontinuous Nonlinear Eigenvalue/Free Boundary Problem 141

and @ and w satisfy the boundary conditions

@ - w = O o n Z

(4*3) QN + w N = o on 2.

By the choice of coordinates ~ ~ ( 0 ) = l/uN(0) > 0; ~ ~ ( 0 ) = uj (0 ) /uN(O) = 0, 1 < j < N - 1. As in [8], the equations (4.1)’ (4.2) are elliptic and the boundary conditions (4.3) are coercive; this shows that T i s an analytic hypersurface.

An alternative proof would use [7], Theorem 3.1: directly; this requires showing that @, ty E C2 (U+ u Z), which may be done as in the remark following [7], Lemma 5.2.

Appendix Expansion of the Green’s function in spherical harmonics

Begin with the fundamental solution for Laplace’s equation

1 ,.2-N Go(‘) = N(2 - N ) ON

is the volume of the unit ball. Then Green’s function for 2 x N / 2

NT(N/2) where wN =

the sphere is

G ~ Y ) = Go(b - Y I ) - Go

It follows that

(A.0) G(r0, U, ro, u’) = GO(r0 fm) - GO(f1 + ri - 2ri cos V )

with cosy = o - 0‘. Now expand the second term on the right in Legendre polynomials ([lo], Lemma 18)

and apply the Addition Theorem for spherical harmonics ([lo], Theorem 2),

with the q,,, an orthonormal basis for the spherical harmonics of degree f in dimension N ; for each I the number of Fm is

( I + N - 3)! (A.3) n(N,I) = (21 + N - 2)

I ! (N - 2)!

Use (A.2) and (A.3) in (A.l) to get

142 R. Alexander

co(l/l + r$ - 2r; cosy) 0 ri' n W, I )

1 Y/m(o) Y ~ m ( ~ ' ) * 1 - c - - (A.4)

(2 - N ) N U , 21 + N - 2 m-1

For the first term on the right in (A.0) we use [lo], Lemma 19, and the Addition Theorem again, giving

Go(Iroo - row' I )

the series converging uniformly in lo - 0'1 2 q > 0. Combining (A.4) and (A.5) gives finally expansion of our kernel in spherical harmonics

G h , 0, ro, 00

References

[l] Alexander , R. K.; Fleishman, B. A.: Perturbation and bifurcation in a free boundary problem. Trans. 26 Conf. Army Maths., ARO Report 81-1, USARO, Research Triangle Park. NC 1981

(21 Chang , K.-C.: On the multiple solutions of the elliptic differential equations with discontinuous nonlinear terms. Scientia Sinica XXI, No. 2 (1978)

[3] Chang, K.-C.: The obstacle problem und partial differential equations with discontinuous nonlinearitis. Comm. Pure Appl. Math. XXXIII, No. 2 (1980) 117-146

[4] Fle i shman , B. A.; Mahar , T. J.: On the existence of classical solutions to an elliptic fret boundary problem. Differential Equations and Applications (W. Eckhaus und E. M. de Jagar, eds.) Amsterdam: North-Holland 1978

[5] Fle i shman , B. A.; Mahar , T. J.: Analytic methods for approximate solution of elliptic free boundary problems. Nonlinear Analysis 1, No. 5 (1977) 561 -569

[6] Gidas, B.; Ni, W.-M.; Nirenberg, L.: Symmetry and related properties via maximum principle. Commun. Math. Phys. 68 (1979) 209-243

[7] Kinderlehrer , D.; Nirenberg, L.; Spruck. J.: Regularityin elliptic freeboundaryproblerns I. J . d'Anal. Math. 34 (1978) 86-119

[8] Kinderlehrer , D.; Stampacchia , G.: An Introduction to Variational Inequalities and their Applications. New York-London: Academic Press 1980

191 Kuiper , H. J.: Eigenvalue problems for noncontinuous operations associated with quasilinear equations. Arch. Rat. Mech. Anal. 53 (1974) 178- 186

[lo] MUller, C.: Spherical Harmonics. Berlin-Heidelberg-New York: Springer 1966. = Lecture Notes in Mathematics Vol. 17

[ll] S t u a r t , C. A.; To land , J. F.: A variational method for boundary value problems with discontinuous nonlinearities. J. London Math. Soc. (2), 21 (1980) 319-328

[12] Temam, R.: A nonlinear eigenvalueproblem: The shape at equilibrium of a confmed plasma. Arch. Rat. Mech. Anal. 60 (1975) 51 -73

Roger Alexander Department of Mathematical Sciences Rmsselaer Polytechnic Institute Troy, NY 12181 (Received June 3, 1981)