Xiaobiao Jia | Searcharxiv

arXiv2022

Hölder estimates and asymptotic behaviors of solutions of some degenerate elliptic equations on half-spaces

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^{2α}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x) +2x_n^α\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 $α>0$ with proper coefficients. Meanwhile, the Hölder estimates up to the boundary will be obtained using rescaling methods.

math.AP↗

arXiv2019

Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces

We prove that any convex viscosity solution of $\det D^2u=1 $ outside a bounded domain of $\mathbb{R}^n_+$ tends to a quadratic polynomial at infinity with rate at least $\frac{x_n}{|x|^{n}}$ if $u$ is a quadratic polynomial on $\{x_n=0\}$ and satisfies $ μ|x|^2\leq u\leq μ^{-1}|x|^2$ as $|x|\rightarrow \infty$ for some $0<μ\leq \frac{1}{2}$.

math.AP↗