The Laplacian with Robin boundary conditions involving signed measures

Abstract

In this work we propose to study the general Robin boundary value problem involving signed smooth measures on an arbitrary domain Ω of ℝ^d. A Kato class of measures is defined to make the associated form (ℰ_μ, ℱ^μ) a closed one. Moreover, the associated operator Δ_μ is a realization of the Laplacian on L²(Ω). In particular, when |μ| is locally infinite everywhere on ∂Ω, Δ_μ is the laplacian with Dirichlet boundary conditions. On the other hand, we will prove that the semigroup (e^-tΔ_μ)_t≥0 is sandwiched between (e^-tΔ_μ+)_t≥0 and (e^-tΔ_-μ-)_t≥0 and we will see that the converse is also true.