The Laplacian with Robin boundary conditions on arbitrary domains. (English) Zbl 1028.31004

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 [W. Arendt and 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 [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 \(p\).


31C15 Potentials and capacities on other spaces
35J25 Boundary value problems for second-order elliptic equations
31C25 Dirichlet forms
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47D03 Groups and semigroups of linear operators
47D07 Markov semigroups and applications to diffusion processes
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