Dirichlet and Neumann boundary conditions: what is in between?
Given an open set \(\Omega\) in \(\mathbb{R}^n\), an admissible measure on \(\partial \Omega\) is a Radon measure \(\mu\) on the Borel \(\sigma\)-field of some open subset \(\Gamma_{\mu}\) of \(\partial \Omega\) which does not charge sets of capacity zero. For any admissible measure \(\mu\) on \(\partial \Omega\), the authors define a realization \(\Delta_{\mu}\) of the Laplacian in \(L^2(\Omega)\) with general Robin boundary conditions and show that \(\Delta_{\mu}\) generates a holomorphic \(C_0\)-semigroup on \(L^2(\Omega)\) which is sandwiched between the Dirichlet Laplacian and the Neumann Laplacian semigroups. Furthermore, the authors show that, under some locality and regularity assumptions, the generator of any sandwiched semigroup is of the form \(\Delta_{\mu}\). The authors also show that if \(D(\Delta_{\mu})\) contains smooth functions, then \(\mu\) is absolutely continuous with respect to the \((n-1)\)-dimensional Hausdorff measure on \(\Omega\).
scientific article; zbMATH DE number 1921151