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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.

##### MSC:
 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
WIAS-TeSCA
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##### References:
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