Regularity and a Liouville theorem for a class of boundary-degenerate second order equations
Analysis of PDEs
2019-12-17 v1
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.
Keywords
Cite
@article{arxiv.1912.07270,
title = {Regularity and a Liouville theorem for a class of boundary-degenerate second order equations},
author = {Brian Weber},
journal= {arXiv preprint arXiv:1912.07270},
year = {2019}
}