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Shape Design for a Class of Degenerate Parabolic Equations with Boundary Point Degeneracy and Its Application to Boundary Observability

Donghui Yang, Jie Zhong

Abstract

We study a class of degenerate parabolic equations with boundary point degeneracy in dimensions N>=2 and investigate the associated boundary observability problem by means of shape design. While one-dimensional degenerate models have been treated in the literature, the genuinely higher-dimensional case remains much more delicate because the degeneracy occurs at a boundary point and the boundary normal trace cannot be extracted directly near the singularity. We approximate the degenerate equation by a family of uniformly parabolic problems on truncated domains obtained by removing a small neighborhood of the degenerate point. Under a geometric condition on the boundary, we establish uniform estimates for the approximate problems, prove convergence to the solution of the original degenerate equation, and identify the convergence of the boundary normal derivatives under additional regularity. We then combine this approximation scheme with a parabolic Carleman estimate for the approximate backward equations and derive a boundary observability inequality for the limiting degenerate equation. In this way, we obtain a higher-dimensional parabolic counterpart of the shape-design program previously developed for degenerate hyperbolic equations.

Shape Design for a Class of Degenerate Parabolic Equations with Boundary Point Degeneracy and Its Application to Boundary Observability

Abstract

We study a class of degenerate parabolic equations with boundary point degeneracy in dimensions N>=2 and investigate the associated boundary observability problem by means of shape design. While one-dimensional degenerate models have been treated in the literature, the genuinely higher-dimensional case remains much more delicate because the degeneracy occurs at a boundary point and the boundary normal trace cannot be extracted directly near the singularity. We approximate the degenerate equation by a family of uniformly parabolic problems on truncated domains obtained by removing a small neighborhood of the degenerate point. Under a geometric condition on the boundary, we establish uniform estimates for the approximate problems, prove convergence to the solution of the original degenerate equation, and identify the convergence of the boundary normal derivatives under additional regularity. We then combine this approximation scheme with a parabolic Carleman estimate for the approximate backward equations and derive a boundary observability inequality for the limiting degenerate equation. In this way, we obtain a higher-dimensional parabolic counterpart of the shape-design program previously developed for degenerate hyperbolic equations.

Paper Structure

This paper contains 12 sections, 15 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solution Spaces
  4. Spectrum
  5. Weak Solutions
  6. Shape Design
  7. Observability on the Boundary
  8. Boundary Observability for Approximate Parabolic Equations
  9. Computation
  10. Estimate
  11. Approximation
  12. Boundary Observability for the Limiting Equation

Key Result

Lemma 2.1

Let $N \geq 2$ and $\alpha \in (0, 1)$. Then, for all $u \in H_0^1(\Omega; w)$, the following inequality holds:

Theorems & Definitions (35)

  • Remark 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • ...and 25 more