Let $s\in(0,2)$ and $u\in C^\infty({\mathbb{R}}^n,[-1,1])$ be a solution of $(-\Delta)^{s/2} u=u-u^3$ in ${\mathbb{R}}^n$, with $\partial_{x_n}u>0$. For which $n$ and $s$ can one conclude that $u$ is necessarily one-dimensional, i.e. that there exist a function $u_0:{\mathbb{R}}\to[-1,1]$ and a direction $\omega\in {\mathbb{S}}^{n-1}$ such that $u(x)=u_0(\omega\cdot x)$ for every $x\in {\mathbb{R}}^n$? ...

The Bernstein property for the $s$-perimeter holds in ${\mathbb{R}}^n$ when global $s$-minimal graphs are necessarily affine. This property holds true when $n\le 3$ for every $s\in(0,1)$ (by [Figalli-Valdinoci, 2017] and [Savin-Valdinoci, 2013]), and also when $n\le8$ and $s$ is sufficiently close to $1$ (by [Figalli-Valdinoci, 2017] and [Caffarelli-Valdinoci, 2013]). ...

In Theorem 1.2 of [Dipierro-Savin-Valdinoci, 2016] it is proved that $s$-minimal graphs are continuous up to the boundary ``coming from inside''. Namely, if $\Omega_0\subset{\mathbb{R}}^{n-1}$ is a domain with boundary of class $C^{1,1}$, $\Omega:=\Omega_0\times{\mathbb{R}}$, $E_0$ is a continuous subgraph in the $n$th direction, then the minimizer $E$ of the $s$-perimeter in $\Omega$ with datum $E_0$ outside $\Omega$ is also a graph, namely $$ E\cap\Omega = \{x_n < v(x')\}$$ for some $v : \Omega_0\to{\mathbb{R}}$ which is uniformly continuous. ...

In Theorem 4.1 of [Caffarelli-Roquejoffre-Savin, 2010] it is shown that if $p$ belongs to the boundary of a set $E$ which minimizes the $s$-perimeter, then the measures of both $B_r(p)\cap E$ and $B_r(p)\setminus E$ are bounded from below by $c r^n$, for a suitable $c>0$ (that is, both the set and its complement have positive densities). In this setting, the constant $c$ depends on $n$ and $s$. ...

In [Dipierro-Savin-Valdinoci, 2017] several examples of sticky boundary behaviors of $s$-minimal surfaces have been exhibited. Namely, differently from the case of classical minimal surfaces, $s$-minimal surfaces may not attain ``continuously'' the boundary data, but rather meet tangentially the boundary of the domain (even when the domain is convex) and stick to it. ...

Are locally Lipschitz $s$-minimal surfaces locally analytic?

In [Figalli-Valdinoci, 2017] it is shown that Lipschitz implies $C^{1,\alpha}$ for any $\alpha < s$. In [Barrios-Figalli-Valdinoci, 2004], it is shown that $C^{1,\alpha}$ with $\alpha>s/2$ implies $C^\infty$. All in all, we know that Lipscitz implies $C^\infty$ (see Theorem 1.1 in [Figalli-Valdinoci, 2017]), but analyticity is open. ...

Is there a critical dimension for which $s$-minimal surfaces are smooth for $s\in (s_0,1)$ and can be singular for $s\in(0,s_0]$, with $s_0\in(0,1)$?

By [Caffarelli-Valdinoci, 2013], it is known that for any $n\in \{2,\dots,7\}$ there exists $s_0(n)\in [0,1)$ such that $s$-minimal surfaces with respect to domains in ${\mathbb{R}}^n$ are smooth for all $s\in(s_0(n),\, 1)$. ...