Poincar{\'e}-Steklov operator and Calder{\'o}n's problem on extension domains

TL;DR

This paper extends Calderón's problem to complex Sobolev extension domains with fractal shapes, introducing a generalized Poincaré-Steklov operator and establishing stability results for inverse conductivity problems in higher dimensions.

Contribution

It generalizes the Poincaré-Steklov operator to non-Lipschitz domains and proves stability of inverse problems for conductivities near the boundary in higher dimensions.

Findings

Stability of the direct problem for bounded conductivities continuous near the boundary.

Boundary stability of the inverse problem for Lipschitz conductivities.

Stability of the inverse problem for W^{2,∞} conductivities constant near the boundary.

Abstract

We consider Calder{\'o}n's problem on a class of Sobolev extension domains containing non-Lipschitz and fractal shapes. We generalize the notion of Poincar{\'e}-Steklov (Dirichlet-to-Neumann) operator for the conductivity problem on such domains. From there, we prove the stability of the direct problem for bounded conductivities continuous near the boundary. Then, we turn to the inverse problem and prove its stability at the boundary for Lipschitz conductivities, which we use to identify such conductivities on the domain from the knowledge of the Poincar{\'e}-Steklov operator. Finally, we prove the stability of the inverse problem on the domain for W^{2,$\infty$} conductivities constant near the boundary. The last two results are valid in dimension n $\ge$ 3.