Hidden Boundary Trace Regularity and an Observability Estimate with Interior Remainder for Boundary-Degenerate Hyperbolic Equations

TL;DR

This paper investigates boundary trace regularity and observability for boundary-degenerate hyperbolic equations, establishing well-posedness, trace estimates, and a large-time observability estimate with interior remainder, highlighting a critical threshold obstacle.

Contribution

It introduces a framework for analyzing boundary trace regularity and observability in degenerate hyperbolic equations, including new estimates and identification of a critical degeneracy threshold.

Findings

Established well-posedness in weighted Sobolev spaces.

Proved an $L^2$ trace estimate for the normal derivative.

Derived a large-time observability estimate with interior remainder.

Abstract

We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\mcA\vp=-\Div(A\nabla \vp)$, where $A=\diag(1,r^\al)$ and $\al\in(0,1)$. We establish well-posedness in weighted Sobolev spaces and prove an $L^2$ trace estimate for the normal derivative on the nondegenerate side $r=1$. Using truncated geometries and Carleman weights adapted to the anisotropic degeneracy, we derive a large-time observability estimate with a lower-order interior remainder. We also identify a framework-level obstruction at the critical threshold $\al=1$: the weighted Dirichlet coercivity underlying the subcritical analysis loses uniformity and exhibits a logarithmic loss on truncated domains.