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

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 $x_0$.