Optimal Sobolev Regularity for Linear Second-Order Divergence Elliptic Operators Occurring in Real-World Problems

摘要: 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 $\partial \Omega \setminus \Gamma$ $\Gamma \subset \partial \Omega$, respectively, discontinuous coefficient functions $\mu$. We develop a general geometric framework for $\Omega$, $\Gamma$, $\mu$ in which it is possible to prove that \nabla +1$ provides an isomorphism from $W^{1,q}_\Gamma(\Omega)$ $W^{-1,q}_\Gamma(\Omega)$ some $q>3$. indicate relevant examples real-world applications.