By Günter Schwarz
Hodge thought is a typical instrument in characterizing range- ential complexes and the topology of manifolds. This e-book is a learn of the Hodge-Kodaira and similar decompositions on manifolds with boundary below customarily analytic points. It goals at constructing a style for fixing boundary price difficulties. Analysing a Dirichlet shape at the external algebra package permits to provide a cultured model of the classical decomposition result of Morrey. A projection method ends up in life and regularity theorems for a large classification of boundary worth difficulties for differential varieties and vector fields. The ebook hyperlinks elements of the geometry of manifolds with the idea of partial differential equations. it really is meant to be understandable for graduate scholars and mathematicians operating in both of those fields.
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Extra info for Hodge Decomposition—A Method for Solving Boundary Value Problems
Example text
T h e r e are several m e t h o d s to establish t h e ellipticity of b o u n d a r y value problems. A m o n g others, t h e r e is t h e Lopatinskii-Sapiro condition which provides us with a direct c o m p u t a t i o n a l criterion. Here we f o r m u l a t e this condition for vector bundle valued b o u n d a r y value problems, following t h e p r e s e n t a t i o n of H S r m a n d e r [85]. ) A m o n g o t h e r features, elliptic o p e r a t o r s are F r e d h o l m a n d s m o o t h i n g .
E. the space of all bounded linear functionals ~" : ~ --* ~ . l N in//3 to be "weakly convergent" to x E ~ , if ~ ' ( x j - x) j ~ oo ~0 V9v E 2 B * and write x j ~ x . Obviously the weak limit is unique. The uniform boundedness principle (BanachSteinhaus theorem), cf. Yosida [72], implies that each bounded sequence (xj)jenv has a weakly convergent subsequence (Xjk)ke~V such that xjk ~ x. In turn, each weakly convergent sequence is bounded. Let A : B 1 ~ J~2 be a linear operator. Its dual A' : IB~ ~ ~ is defined by (A'~)(x) = ~(Ax) Vx ~ B1 If A is bounded on B 1 , then A' is a bounded linear operator too.
E. for all x E ~b(Ub ) and all v E 5/'~, the principal symbol Psym(A)(v,x) : F --~ F is an isomorphism, iff v ~ O. (ii) The boundary conditions Bj are elliptic in the sense that for ali x E ~b(Ub) and a11 ~ E 5/n--X\{o}, the map M +X~V - , GFj l ~j