Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

TL;DR

This paper establishes Carleman estimates for anisotropic elliptic operators with jumps across an interface, identifying necessary geometric conditions for the weight function to ensure the estimates hold.

Contribution

It introduces new Carleman estimates for anisotropic elliptic operators with interface jumps and determines the sharp geometric conditions required.

Findings

Existence of a suitable weight function for Carleman estimates.

Necessary conditions on the weight function are identified.

Results apply to anisotropic diffusion matrices with jumps.

Abstract

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.