Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations (Q2842805)

Consider the following fully nonlinear elliptic equation NEWLINE\[NEWLINE\begin{cases} F(D^2u,Du,u,x)=0\text{ in }\Omega=\Omega_0\setminus \Omega_1 \\ u|_{\partial \Omega_0}=0\text{ and } u|_{\partial \Omega_1}=1 \end{cases}\tag{1}NEWLINE\]NEWLINE where \(\Omega\) is a convex ring domain, i.e., \(\Omega_1\subset \subset \Omega_0\) are convex regions in \(\mathbb R^n\).NEWLINENEWLINEThe main result of the paper is a global geometric lower bound for the second fundamental form of the level surfaces of the solutions to the equation under appropriate conditions on the boundary geometry and the structure of the function \(F\). More specifically, for a function \(u\) defined on \(\Omega\), denote the level surface by \(\sum^c=\{x\in \bar\Omega: u(x)=c\}\) and denote the smallest principal curvature of \(\sum^c\) at \(x\) by \(\kappa_s(x)\). Let \(\kappa^c=\inf_{x\in \sum^c}\kappa_s(x)\). It is shown that for a classical solution \(u\) of equation (1), a global lower bound for \(\kappa^c\) can be established.