MATH 8141, PDES

FALL 2010, C. E. GUTIERREZ

PRELIMINARIES

NOTATIONGiven 1, ..., n non-negative integers we set = (1, ..., n), ! = 1!...n!, || =

1 + ... + n, D =||

x11 ...xnn. Given a complex-valued function f(x), the support of

f is the closure of the set {x : f(x) 6= 0}. Given f and g measurable, complex-valuedfunctions defined over Rn the convolution of f and g is the function defined by

f g(x) =Rn

f(x y)g(y) dy.

Given an open subset of Rn the following classes are defined:

CN ()

= {f : C; f and all its derivatives up the order N are continuous functions in },C()

= {f : C; f and all its derivatives of any order are continuous functions in },C0 () = {f : C; f C() and supp(f) is a compact subset of },S = {f : Rn C; f C(Rn), and sup

xRn|xDf(x)|

2 8141, PDES, FALL 2010, C. E. GUTIERREZ

2. Leibnizs Formula.Let P (x1, ..., xn) be a polynomial in n variables with complex coefficients. If u(x) and

v(x) are C-functions (or sufficiently smooth) we have

P (D)(uv)(x) =

P ()(D)u(x)!

Dv(x).

In particular

D(uv)(x) =

!!( )!D

u(x)Dv(x).

To prove this formula we may note the fact that

P (D)ex = P ()ex,

where x = ni=1 xii.3. The Inverse Function Theorem.Let 1 Rn, 2 Rm, open sets, and u : 1 2. We say that the function u is a

Ck-diffeomorphism if u is bijective, u Ck(1), and its inverse function u1 Ck(2).Let Rn open, a , and f : Rn is function such that f Ck(), f(x) =

(f1(x), ..., fn(x)). If the Jacobian matrix of f is the n n matrix given by

Jf (x) =(fixj

(x))ij

If the Jacobian determinant

det(fixj

(a))6= 0,

then there exist a neighborhood U of a and a neighborhood V of b = f(a) such thatf/U : U V is a Ck-diffeomorphism. In addition, if f1 denotes the inverse function off , then the first derivatives of f1 are given by

Jf1(f(x)) = Jf (x)1.

4. Implicit Function Theorem.Let Rn Rm an open set, and let f : Rm be a Ck-function in , f(x, y) =

(f1(x, y), ..., fm(x, y)), x Rn and y Rm. Let (a, b) such thatf(a, b) = 0,

and suppose that the Jacobian determinant

det(fiyj

(a, b))6= 0.

8141, PDES, FALL 2010, C. E. GUTIERREZ 3

Then there exist a neighborhood U of a, a neighborhood V of b, and a function g : U V ,g Ck(U) such that

g(a) = b,

andf(x, g(x)) = 0, x U.

5. The Divergence Theorem.Let Rn be a bounded, open and connected set such that its boundary is C1. Let

denote the unit outward normal to . Given any vector field W (x) = (w1(x), ..., wn(x))which is C1(), we then have

divW dx =

W d(x).

6. Differentiation under the integral sign.Let Rn measurable, and x0 Rn. Suppose that f(x, t), x, t Rn is a function

such thatf(x, ) L1(), for |x x0| < ,

and the partial derivativef

xj(x, t) exists and there exists g(t) L1() such that fxj (x, t)

g(t),for t and |x x0| < .

Then the functionF (x) =

f(x, t) dt

is differentiable with respect to xj in andF

xj(x) =

f

xj(x, t) dt.

7. Minkowskys inequality.Let 1 p

4 8141, PDES, FALL 2010, C. E. GUTIERREZ

9. Youngs inequality.Let 1 p , f L1(Rn) and g Lp(Rn). Then

f gp f1gp.

10. Let 1 p , f Lp(Rn) and g Lq(Rn) with 1p

+1q

= 1. Then

f g fpgq.

11. Let f be a function defined over Rn, h Rn, and the translation operator hf(x) =f(x h). If 1 p