Interior pointwise $C^{1}$ and $C^{1,1}$ regularity of solutions for general semilinear elliptic equation in nondivergence form

TL;DR

This paper establishes $C^{1}$ and $C^{1,1}$ regularity results for solutions of general semilinear elliptic equations in nondivergence form, under weaker assumptions on the data and coefficients than previously known.

Contribution

It extends regularity results to equations with more general nonlinear terms $f(x,u)$ and weaker coefficient regularity conditions, including optimal Dini continuity assumptions.

Findings

Solutions are $C^{1}$ under small modulus $C^{1,1}$ Newtonian potential conditions.

Solutions are $C^{1,1}$ if coefficients are Dini continuous at a point.

Regularity results hold for unbounded coefficients with specific integrability conditions.

Abstract

In this paper, we obtain $C^{1}$ and $C^{1,1}$ regularity of $L^{n}$-viscosity solutions for general semilinear elliptic equation in nondivergence form under some more weaker assumptions, which generalize the result for equations with nonhomogeneous term $f(x)$ to $f(x,u)$. In particular, the nonhomogeneous term $f(x,u)$ is assumed optimally to satisfy unform Dini continuity condition in $u$ and modified $C^{1,1}$ Newtonian potential condition in $x$. For unbounded coefficients, if $a_{ij}$ is $C_{n}^{-1,1}$ at $x_{0}\in\Omega$ with small modulus, $b_{i}\in L^{q}(\Omega)$ for some $q>n$, the solution is $C^{1}$ at $x_{0}$. Furthermore, if $a_{ij},~b_{i}$ are Dini continuous at $x_{0}$, the solution is $C^{1,1}$ at $x_{0}$.