Stickiness property of $s$-minimal surfaces
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Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci59
Cite as: P-180702.5
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 [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.
Is it possible to describe the stickiness phenomenon in a suitable ``generic'' class? How special is the case in which the $s$-minimal surfaces meet transversally the boundary (as it occurs for halfspaces)?
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Created at: 2018-07-02 18:09:47Z
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