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

### Gaston Beltritti

Universidad Nacional de Río Cuarto, Argentina### Julio D. Rossi

Universidad de Buenos Aires, Argentina

## Abstract

We deal with existence and uniqueness for continuous solutions to the equation $u(x,t)=med_{(y,s)∈J(x,t)}u(y,s)$ for $(x,t)∈R_{n}×(0,∞)$ with a prescribed datum $u(x,t)=f(x)$ for $(x,t)∈R_{n}×(−∞,0]$. Here $med_{(y,s)∈J(x,t)}u(y,s)$ stands for the median value of $u$ in the set $J(x,t)=B(x,R)×[t−(δ+γ),t−δ]$. In addition we show that when we consider the family of sets $J_{r}(x,t):=B(x,rR)×[t−r_{2}(δ+γ),t−r_{2}δ]$ then the corresponding solutions $u_{r}$ converge as $r→0$ 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 $∇u(x,t)$ 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.

## Cite this article

Gaston Beltritti, Julio D. Rossi, Nonlocal Averages in Space and Time Given by Medians and the Mean Curvature Flow. Z. Anal. Anwend. 39 (2020), no. 2, pp. 223–243

DOI 10.4171/ZAA/1658