Non-homogeneous boundary value problems for second-order degenerate hyperbolic equations and their application

TL;DR

This paper develops a new theoretical framework for second-order degenerate hyperbolic equations with non-homogeneous boundary conditions, establishing existence, regularity, and controllability results in weighted Sobolev spaces.

Contribution

It introduces a Dirichlet map for degenerate elliptic operators and extends classical solution theories to degenerate hyperbolic equations with low regularity boundary data.

Findings

Established existence and regularity of weak solutions in weighted Sobolev spaces.

Derived energy estimates and well-posedness for low regularity boundary inputs.

Proved an approximate controllability criterion generalizing the Hilbert Uniqueness Method.

Abstract

We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the degenerate weight. A Dirichlet map is constructed for the degenerate elliptic operator, leading to a solution theory that extends classical approaches to the degenerate setting. In particular, we derive energy estimates and well-posedness for boundary inputs of low regularity (in appropriate trace spaces), even though the classical Dirichlet-to-Neumann framework is not directly applicable in the degenerate setting. As an application, we prove an approximate controllability criterion, which generalizes the Hilbert Uniqueness Method to degenerate wave equations. Our framework accommodates higher-dimensional degenerate waves, non-homogeneous boundary conditions, and weighted functional analysis. We also illustrate how our criterion connects to higher-dimensional Grushin equations and waves with single-point degeneracy, and we highlight the remaining unique continuation/observability issue as an open problem.