English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The nonlocal $s$-fractional minimal surface equation for $\Sigma= \partial E$ where $E$ is an open set in $R^N$ is given by $$ H_\Sigma^ s (p) := \int_{R^N} \frac {\chi_E(x) - \chi_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in \Sigma. $$ 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 $7$.

Nonlocal $s$-minimal surfaces and Lawson cones