Uniqueness results for inverse Robin problems with bounded coefficient

TL;DR

This paper investigates the uniqueness of the Robin inverse problem with bounded coefficients, establishing that uniqueness holds in two dimensions but may fail in higher dimensions, and discusses related open problems.

Contribution

It provides new results on the dimensional dependence of uniqueness in Robin inverse problems with bounded coefficients and highlights open questions about harmonic gradients.

Findings

Uniqueness holds in 2D for Robin inverse problems.

Uniqueness may not hold in higher dimensions.

Raises open questions on harmonic gradients.

Abstract

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, $r\textgreater{}n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context.