Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates

TL;DR

This paper proves boundary controllability for heat equations with inverse square boundary singularities in multiple dimensions, using a novel global Carleman estimate that handles the singular potential and boundary conditions.

Contribution

It introduces the first boundary controllability result for such parabolic equations with inverse square boundary singularities in more than one dimension, via a new global Carleman estimate.

Findings

Established boundary controllability for parabolic equations with inverse square boundary singularities.

Developed a novel global Carleman estimate combining two inequalities with specialized weights.

Achieved controllability results in all spatial dimensions for critically singular potentials.

Abstract

We consider heat operators on a convex domain $\Omega$, with a critically singular potential that diverges as the inverse square of the distance to the boundary of $\Omega$. We establish a general boundary controllability result for such operators in all dimensions, in particular providing the first such result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the appropriate boundary conditions and the $H^1$-energy for this problem. The estimate is derived by combining two intermediate Carleman inequalities with distinct and carefully constructed weights involving non-smooth powers of the boundary distance.