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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 39, Issue 2, 2020, pp. 223–243
DOI: 10.4171/ZAA/1658

Published online: 2020-04-07

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

Gaston Beltritti and Julio D. Rossi

(1) Universidad Nacional de Río Cuarto, Argentina
(2) Universidad de Buenos Aires, Argentina

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.

Keywords: Nonlocal evolution problems, 1-Laplacian, mean value properties

Beltritti Gaston, Rossi Julio: Nonlocal Averages in Space and Time Given by Medians and the Mean Curvature Flow. Z. Anal. Anwend. 39 (2020), 223-243. doi: 10.4171/ZAA/1658