On Neumann-Poincaré operators and self-adjoint transmission problems

We discuss the self-adjointness in $L^2$-setting of the operators acting as $-\mathrm{\nabla }\cdot h\mathrm{\nabla }$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $\mathrm{\Sigma }$, 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 $\mathrm{\Sigma }$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $\mathrm{\Sigma }$, which is new for the Lipschitz setting.