Hardy inequality and Pohozaev identity for operators with boundary singularities: some applications

Abstract

We consider the Schr\"{o}dinger operator $A_\l:=-\D -\l/|x|^2$, $\l\in \rr$, when the singularity is located on the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$ The aim of this Note is two folded. Firstly, we justify the extension of the classical Pohozaev identity for the Laplacian to this case. The problem we address is very much related to Hardy-Poincar\'{e} inequalities with boundary singularities. Secondly, the new Pohozaev identity allows to develop the multiplier method for the wave and the Schr\"{o}dinger equations. In this way we extend to the case of boundary singularities well known observability and control properties for the classical wave and Schr\"{o}dinger equations when the singularity is placed in the interior of the domain (Vanconstenoble and Zuazua \cite{judith}).