Non-homogeneous boundary value problems for second-order degenerate hyperbolic equations and their application
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Yang, Donghui, Zhong, Jie
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7128514428
Cites:
Cited by:
Related to: