Carleman estimates and observability inequalities for parabolic equations with interior degeneracy

TL;DR

This paper develops Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, including cases where degeneracy occurs within the control region, addressing a previously unstudied scenario.

Contribution

It introduces novel Carleman estimates for parabolic equations with interior degeneracy within the control region, advancing controllability theory in this context.

Findings

Established new Carleman estimates for interior degeneracy

Proved controllability results for parabolic equations with interior degeneracy

Extended previous methods to handle degeneracy within control regions

Abstract

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on controllability results through Carleman estimates for the associated adjoint problem. The novelty of the present paper is that the degeneracy is also in the interior of the control region, so that no previous result can be adapted to this situation.