Liouville theorems for fully nonlinear elliptic equations on half spaces

TL;DR

This work establishes two Liouville-type theorems for viscosity solutions of fully nonlinear uniformly elliptic equations on half-spaces: a first-order linear profile and a second-order quadratic profile. The authors introduce a short, self-contained approach based on boundary pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity, together with Carleson-type and Hopf-type estimates, to control the solution’s growth via scaling. The results hold under minimal operator assumptions (uniform ellipticity) for equations of the form $F(D^2u)=0$ and $F(D^2u)=1$, leveraging boundary regularity rather than global growth assumptions. This advances Liouville-type theory for fully nonlinear equations on domains with boundaries and highlights the role of boundary regularity in deriving asymptotic profiles.

Abstract

In this note, we prove two Liouville theorems for fully nonlinear uniformly elliptic equations on half spaces. The main tools are the boundary pointwise regularity, the Hopf type estimate and the Carleson type estimate. Our new proof is rather short.

Theorems & Definitions (15)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1: Boundary pointwise $C^{1,\alpha}$ regularity
• Lemma 2.2: Boundary pointwise $C^{2,\alpha}$ regularity
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5: Carleson type estimate