Nonlocal $s$-minimal surfaces and Lawson cones

Abstract

The nonlocal $s$-fractional minimal surface equation for $\Sigma =\partial E$ where $E$ is an open set in $R^N$ is given by $$H_{\Sigma }^s(p):={\int }_{R^N}\frac{{\chi }_E(x)-{\chi }_{E^c}(x)}{|x-p{|}^{N+s}}dx=0\text{for all }p\in \Sigma .$$ Here $0<s<1$, $\chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s\to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s-$minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $R^3$, the {\bf nonlocal catenoid}, an analog of the standard catenoid $|x_3|=\log (r+\sqrt{r^2-1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $|x_3|=r\sqrt{1-s}$. We also find a two-sheet embedded $s$-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any $0<s<1$, $n,m\ge 1$, $s-$minimal Lawson cones $|v|=\alpha |u|$, $(u,v)\in R^n\times R^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n+m=7$, which suggests that unlike the classical theory (or the case $s$ close to 1), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.

92 pages, 2 figures

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• Nonlocal diffusion and applications