## The Dirichlet Problem with L2-Boundary Data for Elliptic Linear EquationsThe 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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### Contents

INTRODUCTION _ | 1 |

Weighted Sobolev space W12 | 7 |

with L2boundary data | 142 |

Copyright | |

1 other sections not shown

### Other editions - View all

The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations Jan Chabrowski Limited preview - 2006 |

The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations Jan Chabrowski No preview available - 2014 |

### Common terms and phrases

absolutely continuous assume assumption boundary condition boundary data boundary dQ bounded domain Carleson measure Chapter class C2 coefficients completes the proof Consequently constant independent construct continuous function cubes define denote Dini condition Dirichlet problem dx Jq Jq elliptic equations elliptic operator energy estimate equation Lu exists a constant exists a positive finite G L2(dQ G Rn harmonic measure Hausdorff measure Holder inequality Jq Jq Jq Jq-q JQi JQi JRt JRt L2-boundary data Lebesgue measure Lemma Lipschitz Lipschitz continuous mapping Moreover norm observe obtain obvious partition of unity point of dQ positive constant problem with L2-boundary proof of Theorem prove regularized distance Rn-i satisfies Section sequence Sobolev inequality solution in W1,2(Q solution of 2.5 solution u G solve the Dirichlet subset sufficiently small supp Suppose test function Theorem 3.1 u2 dSx uniformly unique solution weighted Sobolev space Wlo'c Q Young's inequality