Appendix K. Implicit Function Theorem
I can endure no more, I demand you remember who you are.- From "I Need a Doctor" by Dr. Dre featuring Eminem and Skylar Grey
In this appendix we collect results that are used in various places in the book.
We discuss the implicit function theorem, Holder and Sobolev spaces, harmonic
maps, and the spectrum of the Hodge-de Rham Laplacian acting on differential
forms on the unit sphere sn.
1. The implicit function theorem
In this section we recall some basic results in functional analysis related to the
applications of the implicit function theorem (IFT) discussed in Chapters 33 and
34.
All of the Banach and Hilbert spaces in this section are real.
1.1. The Lax-Milgram-Lions theorem.
We used the following result to find weak solutions to the boundary value
problem in Lemma 34.14.
THEOREM K.1 (Lax- Milgram- Lions). Suppose that A is a Hilbert space, B is
a normed linear space, and f : Ax B ---+JR. is a continuous bilinear functional. Then
the following two statements are equivalent:
(1) (Coercive) There exists a constant c > 0 such that
inf sup If (a, b)I 2: c.
llbll=^1 llall:51
(2) (Existence of a weak inverse) For every continuous linear functional L :
B ---+ JR. there exists aL E A such thatf(aL,b)=L(b)
for all b EB.
Note that if A= B and if there exists c > 0 such that f (a, a) 2: c llall~, then
f is coercive.
1.2. Contraction mapping theorem.
The following contraction mapping theorem is useful in proving existence re-
sults.
LEMMA K.2 (Contraction maps have unique fixed points). Let (X, d) be a
complete metric space. If f : X ---+ X is such that there exists A E [O, 1) such that
d (f (x), f (y)) ~.Ad (x, y) for all x, y EX, then there exists a unique point x 00 EX
for which f (xoo) = Xoo·
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