The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization Δμ of the Laplacian on L2(Ω) with generalized Robin boundary conditions where Ω is an arbitrary open subset of Rn and μ is a Borel measure on the boundary ∂Ω of Ω. This operator Δμ generates a sub-Markovian C0-semigroup on L2(Ω). If dμ=β dσ where β is a strictly positive bounded Borel measurable function defined on the boundary ∂Ω and σ the (n−1)-dimensional Hausdorff measure on ∂Ω, we show that the semigroup generated by the Laplacian with Robin boundary conditions Δβ has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in Lp(Ω) is independent of p∈[1,∞). Our approach constitutes an alternative way to Daners who considers the (n−1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.