Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

Abstract

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on ${\mathbb{R}}^n$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain $H^2({\mathbb{R}}^n)$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, $\delta$ and ${\delta }^{\prime }$-type, assigned either on a $n-1$ dimensional compact boundary $\Gamma =\partial \Omega$ or on a relatively open part $\Sigma \subset \Gamma$. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

Final revised version, to appear in Journal of Differential Equations

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