H\"older estimates and asymptotic behaviors of solutions of some degenerate elliptic equations on half-spaces

TL;DR

This paper investigates the asymptotic behavior and boundary regularity of viscosity solutions to certain degenerate elliptic equations in half-spaces, establishing H"older estimates and analyzing solution decay near boundaries.

Contribution

It provides new H"older boundary estimates and characterizes the asymptotic behavior of solutions to a class of degenerate elliptic equations with specific degeneracy.

Findings

Established boundary H"older estimates for solutions.

Described asymptotic decay rates of solutions near the boundary.

Analyzed the influence of degeneracy parameter lpha on solution behavior.

Abstract

In this paper we report the asymptotic behaviors of viscosity solutions of the following degenerate elliptic equations \begin{equation*}\label{main-Eq} Lu=x_n^{2\alpha}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x) +2x_n^{\alpha}\sum_{i=1}^{n-1}a_{in}(x)D_{in}u(x)+D_{nn}u(x)=0 \end{equation*} outside a bounded domain of the upper half space for $\alpha>0$ with proper coefficients. Meanwhile, the H\"older estimates up to the boundary will be obtained using rescaling methods.