The Laplacian with Robin Boundary Conditions involving signed measures

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.