The Laplacian with Robin Boundary Conditions involving signed measures

TL;DR

This paper investigates the Laplacian with Robin boundary conditions involving signed measures, establishing conditions for well-posedness and analyzing the associated semigroup's bounds and properties.

Contribution

It introduces a Kato class of measures for Robin boundary problems with signed measures and characterizes the Laplacian operator's realizations and semigroup bounds.

Findings

Defined a Kato class for signed measures ensuring form closability

Characterized the Laplacian with Robin boundary conditions involving signed measures

Proved bounds for the associated semigroup and their equivalence

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.