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Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy

Qing Liu, Erbol Zhanpeisov

TL;DR

This work proves Liouville-type nonexistence results for fully nonlinear elliptic and parabolic equations in which ellipticity degenerates at the boundary. By linking viscosity-solution techniques with implicit boundary conditions generated by boundary degeneracy, the authors show that bounded solutions must be identically zero under natural structural assumptions, with stronger conclusions available when additional regularity (via condition $(F5)$) holds. The results include both boundary-continuous and interior-bounded solution frameworks, and extend to parabolic equations where a dynamic boundary condition emerges from the degeneracy. Overall, the paper highlights how boundary degeneracy enforces boundary behavior and yields strong uniqueness properties for a broad class of nonlinear degenerate PDEs, including classical operators like the Laplacian, Pucci, and viscous Hamilton–Jacobi–Isaacs operators, in both elliptic and parabolic settings.

Abstract

We study a class of fully nonlinear boundary-degenerate elliptic equations, for which we prove that u \equiv 0 is the only solution. Although no boundary conditions are posed together with the equations, we show that the operator degeneracy actually generates an implicit boundary condition. Under appropriate assumptions on the degeneracy rate and regularity of the operator, we then prove that there exist no bounded solutions other than the trivial one. Our method is based on the arguments for uniqueness of viscosity solutions to state constraint problems for Hamilton-Jacobi equations. We obtain similar results for fully nonlinear degenerate parabolic equations. Several concrete examples of equations that satisfy the assumptions are also given.

Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy

TL;DR

This work proves Liouville-type nonexistence results for fully nonlinear elliptic and parabolic equations in which ellipticity degenerates at the boundary. By linking viscosity-solution techniques with implicit boundary conditions generated by boundary degeneracy, the authors show that bounded solutions must be identically zero under natural structural assumptions, with stronger conclusions available when additional regularity (via condition ) holds. The results include both boundary-continuous and interior-bounded solution frameworks, and extend to parabolic equations where a dynamic boundary condition emerges from the degeneracy. Overall, the paper highlights how boundary degeneracy enforces boundary behavior and yields strong uniqueness properties for a broad class of nonlinear degenerate PDEs, including classical operators like the Laplacian, Pucci, and viscous Hamilton–Jacobi–Isaacs operators, in both elliptic and parabolic settings.

Abstract

We study a class of fully nonlinear boundary-degenerate elliptic equations, for which we prove that u \equiv 0 is the only solution. Although no boundary conditions are posed together with the equations, we show that the operator degeneracy actually generates an implicit boundary condition. Under appropriate assumptions on the degeneracy rate and regularity of the operator, we then prove that there exist no bounded solutions other than the trivial one. Our method is based on the arguments for uniqueness of viscosity solutions to state constraint problems for Hamilton-Jacobi equations. We obtain similar results for fully nonlinear degenerate parabolic equations. Several concrete examples of equations that satisfy the assumptions are also given.
Paper Structure (6 sections, 8 theorems, 78 equations)

This paper contains 6 sections, 8 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Implicit boundary conditions
  3. Liouville-type theorems
  4. Nonexistence of nontrivial bounded solutions in $C(\overline{\Omega})$
  5. Nonexistence of nontrivial bounded solutions in $C(\Omega)$
  6. Generalization for parabolic equations

Key Result

Theorem 1.1

Let $\Omega\subset {\mathbb R}^n$ be a domain satisfying (A). Assume that (F1)--(F4) hold. If $u\in C(\Omega)$ is a viscosity solution to nonlinear eq that can be continuously extended to $\partial \Omega$, then $u\equiv 0$ in $\Omega$.

Theorems & Definitions (17)

  • Theorem 1.1: Liouville-type theorem for boundary continuous solutions
  • Theorem 1.2: Liouville-type theorem for continuous solutions
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Implicit boundary conditions
  • proof
  • Remark 2.4
  • Proposition 3.1: Homogeneous Dirichlet condition
  • proof : Proof of Theorem \ref{['thm main']}
  • Remark 3.2
  • ...and 7 more