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

TL;DR

This paper develops a shape design method for higher-dimensional degenerate parabolic equations with boundary point degeneracy, establishing boundary observability through approximation, uniform estimates, and Carleman estimates.

Contribution

It extends the shape design and boundary observability framework from hyperbolic to higher-dimensional degenerate parabolic equations with boundary degeneracy.

Findings

Established uniform estimates for approximate problems

Proved convergence to the degenerate equation solution

Derived boundary observability inequality for the degenerate case

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.