Continuity properties of Neumann-to-Dirichlet maps with respect to the $H$-convergence of the coefficient matrices

TL;DR

This paper studies how the Neumann-to-Dirichlet boundary operator behaves continuously when the coefficient matrices of elliptic equations change, emphasizing the role of $H$-convergence for nonsmooth coefficients and exploring existence of solutions in inverse conductivity problems.

Contribution

It establishes the continuity of boundary operators with respect to $H$-convergence and proves existence results for variational problems in inverse conductivity, highlighting differences between isotropic and anisotropic cases.

Findings

Continuity of boundary operators under $H$-convergence for nonsmooth coefficients.

Existence of minimizers in anisotropic inverse conductivity problems.

Potential non-existence of minimizers in isotropic cases.

Abstract

We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of convergence is the one provided by $H$-convergence (or $G$-convergence for symmetric matrices). We prove existence results for minimum problems associated to variational methods used to solve the so-called inverse conductivity problem, at least if we allow the conductivities to be anisotropic. In the case of isotropic conductivities we show that on certain occasions existence of a minimizer may fail.