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 $\Omega$ of ${\mathbb{R}}^d$. A Kato class of measures is defined to insure the closability of the associated form $(\mem,\mfm)$. Moreover, the associated operator ${\Delta }_{\mu }$ is a realization of the Laplacian on $L^2(\Omega )$. In particular, when $|\mu |$ is locally infinite everywhere on $\po$, ${\Delta }_{\mu }$ is the laplacian with Dirichlet boundary conditions. On the other hand, we will prove that he semigroup $(\emu)_{t\geq 0}$ is sandwitched between $(\emup)_{t\geq 0}$ and $(\emun)_{t\geq 0}$ and we will see that the converse is also true.