Elliptic Boundary Value Problems on Corner Domain

Monique Dauge9780387501697, 0-387-50169-X

This research monograph focusses on a large class of variational elliptic problems with mixed boundary conditions on domains with various corner singularities, edges, polyhedral vertices, cracks, slits. In a natural functional framework (ordinary Sobolev Hilbert spaces) Fredholm and semi-Fredholm properties of induced operators are completely characterized. By specially choosing the classes of operators and domains and the functional spaces used, precise and general results may be obtained on the smoothness and asymptotics of solutions. A new type of characteristic condition is introduced which involves the spectrum of associated operator pencils and some ideals of polynomials satisfying some boundary conditions on cones. The methods involve many perturbation arguments and a new use of Mellin transform. Basic knowledge about BVP on smooth domains in Sobolev spaces is the main prerequisite to the understanding of this book. Readers interested in the general theory of corner domains will find here a new basic theory (new approaches and results) as well as a synthesis of many already known results; those who need regularity conditions and descriptions of singularities for numerical analysis will find precise statements and also a means to obtain further one in many explicit situtations.

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Table of contents :
front-matter……Page 1
01Introduction……Page 9
02Preliminaries……Page 15
03Fredholm and semi-Fredholm results……Page 32
04Proofs……Page 64
05Two-dimensional domains……Page 110
06Singularities along the edges……Page 134
07Laplace operator……Page 159
08Variational boundary value problems on smooth domains……Page 177
09Variational boundary value problems on polyhedral domains……Page 191
back-matter……Page 218