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

2019/12/16 by Brian Weber, Weber, Brian · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1912.07270

openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Cited by

• Boundary Estimates for the Monge-Ampère Equation in the Polygons with Guillemin Boundary Conditions

Related

• Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy
• C^{1, α} Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions
• Boundary regularity for subelliptic equations in the Heisenberg group
• Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations
• Boundary regularity for a degenerate elliptic equation with mixed\n boundary conditions