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

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.