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OverviewThe general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz subdomains of Riemannian manifolds. In the first part it develops a theory for Cauchy type operators on Lipschitz submanifolds of codimension one (focused on boundedness properties and jump relations). The solution is represented in the form of layer potentials and optimal nontangential maximal function estimates are established. This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, the authors further specialize this discussion to the case of Hodge Laplacian. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDEs of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately. The main tools are those of PDEs and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry. Full Product DetailsAuthor: Mauris Mitrea , Michael TaylorPublisher: American Mathematical Society Imprint: American Mathematical Society Edition: UK ed. Volume: No. 713 Weight: 0.255kg ISBN: 9780821826591ISBN 10: 082182659 Pages: 120 Publication Date: 28 February 2001 Audience: College/higher education , Professional and scholarly , Undergraduate , Postgraduate, Research & Scholarly Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction Singular integrals on Lipschitz submanifolds of codimension one Estimates on fundamental solutions General second-order strongly elliptic systems The Dirichlet problem for the Hodge Laplacian and related operators Natural boundary problems for the Hodge Laplacian in Lipschitz domains Layer potential operators on Lipschitz domains Rellich type estimates for differential forms Fredholm properties of boundary integral operators on regular spaces Weak extensions of boundary derivative operators Localization arguments and the end of the proof of Theorem 6.2 Harmonic fields on Lipschitz domains The proofs of the Theorems 5.1-5.5 The proofs of the auxiliary lemmas Applications to Maxwell's equations on Lipschitz domains Analysis on Lipschitz manifolds The connection between $d_\partial$ and $d_{\partial\Omega}$ Bibliography.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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