Critical dimension for interior regularity for nonlocal minimal surfaces
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Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci45
Cite as: P-180702.3
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
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 $s\in(s_0(n),\, 1)$.
As a matter of fact, by [Savin-Valdinoci, 2013], $s_0(2)=0$.
By [Dávila-del Pino-Wei, 2018], symmetric cones that are critical points of the $s$-perimeter functional are necessarily flat in dimensions $n\in\{2,\dots,6\}$, for all $s\in(0,1)$, but in dimension $7$ there exists a symmetric nonflat cone which, for small values of $s$, is a stable critical point of the $s$-perimeter functional (i.e. the second variation of the functional has positive sign).
Therefore, two possibilities arise: either this cone in ${\mathbb{R}}^7$ is actually a minimizer for the $s$-perimeter (therefore $s$-minimal surfaces in domains of ${\mathbb{R}}^7$ are smooth for $s$ sufficiently close to $1$ and singular for $s$ sufficiently close to $0$), or this cone would provide an example of stable, but not $s$-minimal, cone.
In addition: either the symmetric cones in dimension $n\in\{2,\dots,6\}$ are fully representative of the regularity properties of the $s$-minimal surfaces (and so all the $s$-minimal surfaces are smooth in $ {\mathbb{R}}^n$ for $n\in\{2,\dots,6\}$) or there exists a singular $s$-minimal surface in ${\mathbb{R}}^n$ for some $n\in\{2,\dots,6\}$ whose blow-up is not a symmetric cone.
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Created at: 2018-07-02 18:09:47Z
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