(Non)uniform density estimates as $s\searrow0$
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Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci64
Cite as: P-180702.6
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
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 $s$.
Is it possible to obtain uniform density estimates as $s\searrow0$, or does the density proportion of the $s$-minimal sets degenerate as $s\searrow0$?
This question is quite related to Theorem 1.4 in [Bucur-Lombardini-Valdinoci], in which it is shown that, for $s$ sufficiently small, ``typical'' exterior data force the $s$-minimal sets (or their complement, according to the density at infinity) to completely fill the domain, unless the boundary accumulates densely at any point of the domain: understanding the density properties of $s$-minimal surfaces as $s\searrow0$ would contribute to understand whether or not the latter accumulation phenomenon really occurs or not.
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Created at: 2018-07-02 18:09:47Z
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