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

Donghui Yang, Jie Zhong

TL;DR

The paper addresses second-order degenerate hyperbolic equations with non-homogeneous Dirichlet boundary data by developing a weighted Sobolev framework for $A=-\operatorname{div}(w\nabla\cdot)$ with $w\in A_2$, and constructing a Dirichlet map to lift boundary traces. It establishes well-posedness and regularity for weak and mild solutions, including non-homogeneous boundary inputs, through a cosine-operator approach and a boundary-control lifting mechanism. A HUM-type approximate controllability criterion is derived, linking controllability to an adjoint observability condition and connecting to Grushin-type and single-point degeneracies, while noting open issues in unique continuation. The results extend classical boundary control methods to degenerate settings, enabling high-dimensional analysis with weighted spaces and providing a structured path toward exact controllability questions in degenerate geometries. Overall, the work contributes a rigorous framework for boundary control of degenerate waves and highlights key open problems in observability and continuation properties in the presence of interior degeneracies.

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.

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

TL;DR

The paper addresses second-order degenerate hyperbolic equations with non-homogeneous Dirichlet boundary data by developing a weighted Sobolev framework for with , and constructing a Dirichlet map to lift boundary traces. It establishes well-posedness and regularity for weak and mild solutions, including non-homogeneous boundary inputs, through a cosine-operator approach and a boundary-control lifting mechanism. A HUM-type approximate controllability criterion is derived, linking controllability to an adjoint observability condition and connecting to Grushin-type and single-point degeneracies, while noting open issues in unique continuation. The results extend classical boundary control methods to degenerate settings, enabling high-dimensional analysis with weighted spaces and providing a structured path toward exact controllability questions in degenerate geometries. Overall, the work contributes a rigorous framework for boundary control of degenerate waves and highlights key open problems in observability and continuation properties in the presence of interior degeneracies.

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.
Paper Structure (11 sections, 17 theorems, 144 equations)

This paper contains 11 sections, 17 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Solution space and well-posedness theory
  3. Weighted Sobolev Spaces and Weak Solutions (Homogeneous Dirichlet Case)
  4. Energy estimates and well-posedness with non-homogeneous Dirichlet boundary input
  5. Mild solution with Dirichlet boundary condition
  6. Mild solution with non-homogeneous boundary condition
  7. Non-homogeneous Dirichlet boundary input and Dirichlet map
  8. Application: Approximate controllability
  9. Example 3.1 (interior single-point degeneracy).
  10. Grushin-type degeneracies and anisotropic models (discussion).
  11. Concluding remarks

Key Result

Lemma 2.1

The spaces $(H^1(\Omega;w),(\cdot,\cdot))_{H^1(\Omega;w)}$, $(H_0^1(\Omega;w),(\cdot,\cdot)_{H^1(\Omega;w)})$, and $(H^2(\Omega;w), (\cdot,\cdot)_{H^2(\Omega;w)})$ are Hilbert spaces.

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Conormal derivative
  • Definition 2.6: Weak solution
  • Theorem 2.7: Existence and regularity
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 38 more