Bernstein property for nonlocal minimal graphs
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Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci61
Cite as: P-180702.8
General Description View the group
Let $\Omega$ be a bounded, open subset of ${\mathbb{R}}^n$ (say, with sufficiently regular boundary) and $s\in(0,1)$. In [Caffarelli-Roqueljoffre-Savin 2010] one investigates the problem of the local minimization of the $s$-perimeter functional $$ {\rm Per}_s(E,\Omega):= I_s(E\cap\Omega, E^c\cap\Omega)+I_s(E\cap \Omega,E^c\cap\Omega^c)+I_s(E\cap\Omega^c,E^c\cap\Omega),$$ where $E^c:={\mathbb{R}}^n\setminus E$, $\Omega^c:={\mathbb{R}}^n\setminus \Omega$ and $$ I_s(A,B):=\iint_{A\times B} \frac{dx\, dy}{|x-y|^{n+s}},\qquad{\mbox{for all $A, B\subseteq{\mathbb{R}}^n$ with $A\cap B=\varnothing$.}}$$ The above local minimization occurs among all competitiors $F\subseteq{\mathbb{R}}^n$ with $F\setminus\Omega=E\setminus\Omega$. Minimizers are often called $s$-minimal surfaces (or nonlocal, or fractional, minimal surfaces). When the minimizer is a graph with respect to a certain direction, it is called an $s$-minimal graph. Minimizers with a cone structure are called $s$-minimal cones.
See e.g. [Cozzi-Figalli, 2017] and [Dipierro-Valdinoci, 2018] for reviews on this subject.
The problems proposed in this list focus on the interior and boundary behaviors of nonlocal minimal surfaces.
Problem's Description
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]).
For which $n$ and $s$ does the Bernstein property hold true?
Does it exist a global $s$-minimal graph which is not affine?
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Created at: 2018-07-02 18:09:47Z
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