Null controllability of n-dimensional parabolic equations degenerated on partial boundary

TL;DR

This paper develops advanced Carleman estimates for high-dimensional degenerate parabolic equations on Lipschitz domains, enabling the analysis of null controllability despite boundary degeneracy and nonsmoothness.

Contribution

It introduces novel Carleman estimates tailored for degenerate high-dimensional parabolic equations with nonsmooth boundaries, facilitating controllability analysis in challenging settings.

Findings

Established new Carleman estimates for degenerate equations

Proved null controllability under boundary degeneracy

Unified previous results and extended to new cases

Abstract

This paper extends the Carleman estimates to high dimensional parabolic equations with highly degenerate symmetric coefficients on a bounded domain of Lipschitz boundary and use these estimates to study the controlla?bility the corresponding equations. Due to the nonsmoothness and degeneracy of boundary, the partial integration by parts in Carleman estimates have no meaning on the degenerate and nonsmooth parts of the boundary. To get around of this difficulty, we construct special weight function, and transform some integral terms in degenerate regions into a non-degenerate ones carefully so that the obtained Carleman estimates can still be used to the controllability problem. Our results includes some well-known works as some special cases as well as some interesting new examples.