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
Nonlocal Averages in Space and Time Given by Medians and the Mean Curvature Flow cover
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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)u(x,t)={\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s) for (x,t)Rn×(0,)(x,t) \in \mathbb R^n\times (0,\infty) with a prescribed datum u(x,t)=f(x)u (x,t)=f(x) for (x,t)Rn×(,0](x,t) \in \mathbb R^n \times (-\infty, 0]. Here med(y,s)J(x,t)u(y,s){\textrm{med}}_{(y,s)\in J(x,t)} \, {u}(y,s) stands for the median value of uu in the set J(x,t)=B(x,R)×[t(δ+γ),tδ]J(x,t)=B(x,R) \times [t -(\delta + \gamma),t -\delta]. In addition we show that when we consider the family of sets Jr(x,t):=B(x,rR)×[tr2(δ+γ),tr2δ]J_r(x,t) := B(x,rR) \times [t - r^2(\delta +\gamma),t - r^2\delta] then the corresponding solutions uru_r converge as r0r\to 0 to the unique viscosity solution to the local degenerate parabolic PDE, ut(x,t)=CVu(x,t)u_t(x,t)=C \triangle_{V}u(x,t), where VV is the orthogonal subspace to u(x,t)\nabla u(x,t) and CC 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