The Laplacian with Robin boundary conditions on arbitrary domains
The authors consider the Laplace operator \(\Delta\) on an open set \(\Omega\) in \(\mathbb{R}^n\) with the Robin boundary condition. They deal with the case of an arbitrary Borel measure \(\mu\) on the boundary \(\partial \Omega\), and show that it is always possible to define a realization \(\Delta_\mu\) of the Laplace operator on \(L^2(\Omega)\) with generalized Robin boundary condition. The operator \(\Delta_\mu\) generates a sub-Markovian semigroup on \(L^2(\Omega)\) which allows Gaussian estimates according to [\textit{W. Arendt} and \textit{A. F. M. ter Elst}, J. Oper. Theory 38, 87-130 (1997; Zbl 0879.35041)]. In the case \(\mu=\beta d\sigma\), corresponding to \(\frac{\partial u}{\partial\nu}+ \beta u =0\) on \(\partial \Omega\), where \(\beta\) is a strictly positive bounded Borel measurable function on \(\partial \Omega\), \ \(\sigma\) is the \((n-1)\)-dimensional Hausdorff measure on \(\partial \Omega\), the authors show that the semigroup has Gaussian estimates with modified exponents. The case of \((n-1)\)-dimensional Hausdorff measure on \(\partial \Omega\) was previously treated in [\textit{D. Daners}, Trans. Am. Math. Soc. 352, 4207-4236 (2000; Zbl 0947.35072)] (for the case of an elliptic second order differential operator). The authors' approach is alternative and is more based on a capacity approach. The authors also show that the spectrum of the Laplace operator with Robin condition in \(L^p(\Omega), 1\leq p <\infty\), is independent of \(p\).
scientific article; zbMATH DE number 1968879
scientific article