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

TL;DR

This paper develops a shape-design approximation method for a class of degenerate hyperbolic equations with boundary degeneracy, proving convergence and establishing observability inequalities under certain geometric conditions.

Contribution

It introduces a novel approximation approach for degenerate hyperbolic equations, enabling analysis of observability and boundary behavior.

Findings

Convergence of regularized solutions to the degenerate problem.

Establishment of an observability inequality under geometric conditions.

Development of a weighted functional framework for degenerate equations.

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.