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

Jussi Behrndt,Jonathan Rohleder

doi:10.1088/1361-6420/ab603d

Jussi Behrndt, Jonathan Rohleder

Open Access

https://doi.org/10.1088/1361-6420/ab603d

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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.

Highlights

Let L be a uniformly elliptic, formally symmetric differential expression of the form n n L=− ∂j ajk∂k +aj ∂j − ∂j aj + a (1.1)j,k=1 j=1 on a possibly unbounded Lipschitz domain Ω
We emphasize that Ω is an unbounded Lipschitz domain without any additional geometric restrictions, and that ω may be a bounded subset of ∂Ω even in the case that ∂Ω is unbounded
The interplay between elliptic differential operators and their corresponding Dirichlet-to-Neumann maps is of particular interest for spectral theory and inverse problems, among them the famous Calderon problem, the multidimensional Gelfand inverse boundary spectral problem, and inverse scattering problems on Riemannian manifolds

Summary

Introduction

Let L be a uniformly elliptic, formally symmetric differential expression of the form n n L=− ∂j ajk∂k +aj ∂j − ∂j aj + a (1.1)j,k=1 j=1 on a possibly unbounded Lipschitz domain Ω. In order to define the Dirichlet-to-Neumann map associated with L on the boundary of the unbounded Lipschitz domain Ω we need the following lemma, which is well known for bounded domains and remains valid in the unbounded case. Speaking Theorem 3.4 states that the knowledge of the Dirichletto-Neumann map M (λ) on a nonempty open subset ω of the boundary ∂Ω for sufficiently many λ determines the Dirichlet operator uniquely up to unitary equivalence.

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