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

Alberto Enciso, Arick Shao, Bruno Vergara

TL;DR

The paper addresses boundary null controllability for a parabolic equation with an inverse-square boundary singular potential on a bounded convex domain, proving results in all dimensions. It introduces a novel global Carleman estimate that simultaneously controls Neumann boundary data and the full $H^1$ energy by using two boundary-defining functions and tailored weights, enabling a boundary observability inequality for the dual problem. Combining this with HUM, it establishes boundary null controllability for $\sigma\in(-\tfrac34,0)$ and lower-order terms, and develops corresponding well-posedness and trace theories for strict and weak solutions. The work extends higher-dimensional controllability results for critically singular parabolic operators and supplies robust tools potentially applicable to more general geometric settings and conformally compact manifolds.

Abstract

We consider heat operators on a convex domain $Ω$, with a critically singular potential that diverges as the inverse square of the distance to the boundary of $Ω$. 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.

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

TL;DR

The paper addresses boundary null controllability for a parabolic equation with an inverse-square boundary singular potential on a bounded convex domain, proving results in all dimensions. It introduces a novel global Carleman estimate that simultaneously controls Neumann boundary data and the full energy by using two boundary-defining functions and tailored weights, enabling a boundary observability inequality for the dual problem. Combining this with HUM, it establishes boundary null controllability for and lower-order terms, and develops corresponding well-posedness and trace theories for strict and weak solutions. The work extends higher-dimensional controllability results for critically singular parabolic operators and supplies robust tools potentially applicable to more general geometric settings and conformally compact manifolds.

Abstract

We consider heat operators on a convex domain , with a critically singular potential that diverges as the inverse square of the distance to the boundary of . 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 -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.
Paper Structure (20 sections, 20 theorems, 193 equations, 1 figure)

This paper contains 20 sections, 20 theorems, 193 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boundary asymptotics
  3. Motivation
  4. Boundary controllability and observability
  5. Global Carleman inequality
  6. Further discussions
  7. Organization of the paper
  8. The Carleman estimate
  9. Construction of boundary defining functions
  10. The global pointwise inequality
  11. The global Carleman estimate
  12. Boundary Observability
  13. Elliptic and Semigroup Theory
  14. Strict Solutions
  15. The Neumann Trace
  16. ...and 5 more sections

Key Result

Theorem 1.6

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded domain, with a convex, connected, $C^4$-boundary $\Gamma$, and fix $\sigma \in ( -\frac{3}{4}, 0 )$. Then, Problem (C) is boundary null controllable in any positive time---for any initial data $v_0 \in L^2(\Omega)$ and any $T > 0$, there exists Dirich

Figures (1)

  • Figure 2.1: The domain $\Omega$ with convex boundary $\Gamma$ is depicted together with balls centered at the critical points $x_1,x_2$ of two good boundary defining functions $y_1, y_2$. In a neighborhood of $\Gamma$, these functions agree with the distance to the boundary $d_\Gamma$.

Theorems & Definitions (62)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10: Global Carleman estimate
  • ...and 52 more