On Neumann-Poincaré operators and self-adjoint transmission problems

Abstract

We discuss the self-adjointness in $L^2$-setting of the operators acting as $-\nabla \cdot h\nabla$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $\hat{I}\pounds$, without explicit assumptions on the sign of $h$. We establish a number of sufficient conditions for the self-adjointness of the operator with $H^s$-regularity for suitable $s\in [1,\frac{3}{2}]$, in terms of the jump value and the regularity and geometric properties of $\hat{I}\pounds$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $\hat{I}\pounds$, which is new for the Lipschitz setting.

In this version, we extended the main result for Sobolev regularity with index $s\in [1,3/2]$ and corrected several typos and added additional references