Regularity and a Liouville theorem for a class of boundary-degenerate second order equations

TL;DR

This paper establishes maximum principles, Harnack inequalities, and Liouville-type theorems for a class of boundary-degenerate elliptic equations, with applications in finance, geometry, and PDE theory.

Contribution

It introduces new regularity results and Liouville theorems for boundary-degenerate elliptic PDEs under minimal regularity assumptions.

Findings

Proved maximum principle and Harnack inequality at degenerate boundary.

Established continuity of solutions under local boundedness.

Derived Liouville theorem constraining entire solutions on the half-plane.

Abstract

We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness, we prove continuity. On globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th order Abreu equation from K\"ahler geometry.