Elliptic Boundary Value Problems on Corner Domains

1. Front Matter

   Pages I-VIII

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2. Introduction

   • Monique Dauge

   Pages 2-7

3. Preliminaries

   • Monique Dauge

   Pages 8-24

4. Fredholm and semi-Fredholm results

   • Monique Dauge

   Pages 26-57

5. Proofs

   • Monique Dauge

   Pages 58-103

6. Two-dimensional domains

   • Monique Dauge

   Pages 104-127

7. Singularities along the edges

   • Monique Dauge

   Pages 128-152

8. Laplace operator

   • Monique Dauge

   Pages 153-170

9. Variational boundary value problems on smooth domains

   • Monique Dauge

   Pages 172-185

10. Variational boundary value problems on polyhedral domains

    • Monique Dauge

    Pages 186-212

11. Back Matter

    Pages 214-261

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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.

• BVP
• Boundary value problem
• Laplace operator
• Smooth function
• Sobolev space
• operator