Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains

TL;DR

This paper proves that partial boundary data for elliptic operators on unbounded Lipschitz domains uniquely determines the operator, even with minimal boundary information and weak coefficient regularity.

Contribution

It establishes uniqueness results for inverse boundary value problems with partial data on unbounded Lipschitz domains, including non-self-adjoint cases.

Findings

Unique determination of elliptic operators from partial boundary data

Results hold for unbounded Lipschitz domains with minimal regularity

Applicable to both self-adjoint and non-self-adjoint operators

Abstract

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expression.