Elliptic Boundary Value Problems on Corner Domains Smoothness and As
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
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1341 Monique Dauge
Elliptic Boundary Value Problems on Corner Domains Smoothness and Asymptotics of Solutions
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1341 Monique Dauge
Elliptic Boundary Value Problems on Corner Domains Smoothness and Asymptotics of Solutions
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author Monique Oauge Unite Associee au CNRS no758 Departernent de Mathematiques et d'informatique 2, rue de la Houssiniere 44072 Nantes Cedex 03, France
Mathematics Subject Classification (1980): Primary: 35J; 47 F Secondary: 58G ISBN 3-540-50169-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50169-X Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Many physical phenomena are described by elliptic boundary value problems let us quote vibrating membranes, elasticity, electrostatics, hydrodynamics for instance. Natural domains are often non-smooth ones or they may be "small perturbations" of such nonregular domains. That is why many people are interested in domains with singularities on their boundaries. In this book, we deal with a great variety of domains : we consider conical singularities of course, but also edges, polyhedral corners, combined with various types of cracks, holes or slits. In order to give precise mathematical results, we need to choose a functional framework. So we decided, therefore to choose ordinary hilbertian Sobolev spaces with real exponents (also called SobolevSlobodeckii spaces). Other choices are possible, but we prefered this one for several reasons that we explain in the introduction . We develop a general theory : first, we characterize different fundamental properties of induced operators, in particular regularity, Fredholm and semi-Fredholm properties, and then we give asymptotics of solutions in the neighborhood of singular points of the boundary. Our results can be applied to specific problems : in such cases, it is often possible to get the characteristic conditions we give more precise. As an example, we do this for the Dirichlet problem associated to the Laplace equation. In another paper, we apply them to the Stokes system. Moreover, the type of statements we get can be adapted to other problems than those we consider here : for instance to non-homogeneous boundary da
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