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 $u(x,t)={\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s)$ 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 ${\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s)$ 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, $u_t(x,t)=C \triangle_{V}u(x,t)$, 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.