Nonlocal Averages in Space and Time Given by Medians and the Mean Curvature Flow

Abstract

We deal with existence and uniqueness for continuous solutions to the equation $u(x,t)={\text{med}}_{(y,s)\in J(x,t)}u(y,s)$ for $(x,t)\in {\mathbb{R}}^n\times (0,\infty )$ with a prescribed datum $u(x,t)=f(x)$ for $(x,t)\in {\mathbb{R}}^n\times (-\infty ,0]$. Here ${\text{med}}_{(y,s)\in J(x,t)}u(y,s)$ stands for the median value of $u$ in the set $J(x,t)=B(x,R)\times [t-(\delta +\gamma ),t-\delta ]$. In addition we show that when we consider the family of sets $J_r(x,t):=B(x,rR)\times [t-r^2(\delta +\gamma ),t-r^2\delta ]$ then the corresponding solutions $u_r$ converge as $r\to 0$ to the unique viscosity solution to the local degenerate parabolic PDE, $u_t(x,t)=C{△}_{V}u(x,t)$, where $V$ is the orthogonal subspace to $\nabla u(x,t)$ and $C$ is a positive constant. This PDE turns out to be the equation that describes the mean curvature flow in its level sets formulation.