Regularity results for elliptic problems with measure
Résumé
In this work, we study the solution of the Laplace equation: \[ -\Delta u =g \delta_\sigma\;\;\;\hbox{ in } Q\subseteq \mathbb{R}^3, \] where \(\delta_\sigma\) is the Dirac mass on a crack \(\sigma\) of \(Q\) and \(g\in L^2(\sigma)\).
First, we discuss the existence and the uniqueness of a solution in \(W^{1,p}(Q)\) for \(p<2\) (due to the Dirac mass, the right-hand side is not in \(H^{-1}(Q)\)). Then, we prove the regularity of the solution and a priori estimates in weighted Sobolev spaces.