Král type removability results for $k$-Hessian equation and $k$-curvature equation

Abstract

We consider some removability problem for solutions to the so-called $k$-Hessian equation and $k$-curvature equation. We prove that if a $C^1$ function $u$ is a generalized solution to $k$-Hessian equation $F_k[u]=g(x,u,Du)$ or $k$-curvature equation $H_k[u]=g(x,u,Du)$ in $\Omega \setminus u^{-1}(E)$ for $E \subset \mathbb{R}$, then it is indeed a generalized solution to the same equation in the whole domain $\Omega$, under some hypotheses on $u$, $g$ and $E$.