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Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems. (English) Zbl 1325.35028
Authors’ abstract: On bounded domains \(\Omega \subset \mathbb{R}^3\), we consider divergence-type operators \(-\nabla \cdot \mu \nabla\), including mixed homogeneous Dirichlet and Neumann boundary conditions on \(\partial \Omega \setminus \Gamma\) and \(\Gamma \subset \partial \Omega\), respectively, and discontinuous coefficient functions \(\mu\). We develop a general geometric framework for \(\Omega,\;\Gamma\), and \(\mu\) in which it is possible to prove that \(-\nabla \cdot \mu \nabla+1\) provides an isomorphism from \(W^{1,q}_\Gamma(\Omega)\) to \(W^{-1,q}_\Gamma(\Omega)\) for some \(q > 3\). We indicate relevant examples from real-world applications.

35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35B65 Smoothness and regularity of solutions to PDEs
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