The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations

Jan Chabrowski (auth.)3540544860, 9783540544869, 0387544860

The Dirichlet problem has a very long history in mathematics and its importance in partial differential equations, harmonic analysis, potential theory and the applied sciences is well-known. In the last decade the Dirichlet problem with L2-boundary data has attracted the attention of several mathematicians. The significant features of this recent research are the use of weighted Sobolev spaces, existence results for elliptic equations under very weak regularity assumptions on coefficients, energy estimates involving L2-norm of a boundary data and the construction of a space larger than the usual Sobolev space W1,2 such that every L2-function on the boundary of a given set is the trace of a suitable element of this space. The book gives a concise account of main aspects of these recent developments and is intended for researchers and graduate students. Some basic knowledge of Sobolev spaces and measure theory is required.

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Table of contents :
Introduction….Pages 1-4
Weighted Sobolev space $$tilde W^{1,2}$$ ….Pages 7-19
The Dirichlet problem in a half-space….Pages 20-45
The Dirichlet problem in a bounded domain….Pages 46-66
Estimates of derivatives….Pages 67-77
Harmonic measure….Pages 78-89
Exceptional sets on the boundary….Pages 90-103
Applications of the L 2 -method….Pages 104-116
Domains with C 1,α -boundary….Pages 117-130
The space C n−1 ( $$bar Q$$ )….Pages 131-141
C n−1 -estimate of the solution of the Dirichlet problem with L 2 -boundary data….Pages 142-167