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In this work we propose to study the general Robin boundary value probleminvolving signed smooth measures on an arbitrary domain $\Omega$ of $\mathbbR^d$. A Kato class of measures is defined to insure the closability of theassociated 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 withDirichlet boundary conditions. On the other hand, we will prove that hesemigroup $(\emu)_{t\geq 0}$ is sandwitched between $(\emup)_{t\geq 0}$ and$(\emun)_{t\geq 0}$ and we will see that the converse is also true.

The Laplacian with Robin Boundary Conditions involving signed measures