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

TL;DR

This paper establishes Liouville-type theorems for fully nonlinear boundary-degenerate elliptic and parabolic equations, showing that under certain conditions, the only bounded solution is trivial, using viscosity solution techniques.

Contribution

It introduces new Liouville-type results for degenerate fully nonlinear equations without explicit boundary conditions, leveraging boundary degeneracy as an implicit boundary condition.

Findings

Uniqueness of the trivial solution for certain degenerate elliptic equations

Extension of results to degenerate parabolic equations

Identification of conditions on degeneracy and regularity for nonexistence of nontrivial solutions

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.