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Shape-Design Approximation for a Class of Degenerate Hyperbolic Equations with a Degenerate Boundary Point and Its Application to Observability

Dong-Hui Yang, Jie Zhong

Abstract

We study a class of degenerate hyperbolic equations in a bounded domain whose degeneracy occurs at a boundary point. We first develop the weighted functional framework, prove well-posedness of the degenerate problem, and establish regularity away from the degenerate point. We then introduce a shape-design approximation obtained by removing a small neighborhood of the degenerate boundary point, which yields uniformly non-degenerate hyperbolic problems on regularized domains. We prove that the regularized solutions converge to the solution of the original degenerate equation, including the convergence of the boundary normal derivatives away from the degenerate point. Finally, under a geometric condition on the observation boundary, we derive an observability inequality for the degenerate equation by combining the uniform observability of the regularized problems with the limit passage.

Shape-Design Approximation for a Class of Degenerate Hyperbolic Equations with a Degenerate Boundary Point and Its Application to Observability

Abstract

We study a class of degenerate hyperbolic equations in a bounded domain whose degeneracy occurs at a boundary point. We first develop the weighted functional framework, prove well-posedness of the degenerate problem, and establish regularity away from the degenerate point. We then introduce a shape-design approximation obtained by removing a small neighborhood of the degenerate boundary point, which yields uniformly non-degenerate hyperbolic problems on regularized domains. We prove that the regularized solutions converge to the solution of the original degenerate equation, including the convergence of the boundary normal derivatives away from the degenerate point. Finally, under a geometric condition on the observation boundary, we derive an observability inequality for the degenerate equation by combining the uniform observability of the regularized problems with the limit passage.
Paper Structure (10 sections, 22 theorems, 238 equations)

This paper contains 10 sections, 22 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Functional Setting and Well-posedness
  3. Solution spaces
  4. Spectrum
  5. Existence of weak solution
  6. Shape-Design Approximation
  7. Observability via Shape Design
  8. Observability for the approximate equations
  9. Approximation
  10. Observability

Key Result

Theorem 1.3

Let $\alpha\in (0,1)$ and $\varepsilon\in (0,\frac{1}{8}R_0)$. Let $y^0\in C_0^\infty(\Omega)$, $y^1\in L^2(\Omega)$, and $f\in L^2(Q)$. Suppose $y$ is the solution of 03.06.m with respect to $(y^0,y^1,f)$, and $y_\varepsilon$ is the solution of 12.12.15 with respect to $(y_\varepsilon^0=y^0, y_\var Moreover, if we additionally assume $y^0\in C_0^\infty(\Omega)$, $y^1\in C_0^\infty(\Omega)$, and $

Theorems & Definitions (50)

  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['08.15.L1']}
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 40 more