Continuity of $s$-minimal graphs ``coming from inside''
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Posted online: 2018-07-02 18:09:47Z by Enrico Valdinoci367
Cite as: P-180702.7
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 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.
That is, the graph of $E$ is continuous ``from the inside'' up to the boundary of the cylinder $\Omega$, but it can exhibit jumps at the boundary.
This inner continuity result is proved under the assumption that $\Omega_0$ above is of class $C^{1,1}$. Is the same result true under weaker regularity? Is it enough, for instance, to suppose just that $\Omega_0$ is Lipschitz?
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Created at: 2018-07-02 18:09:47Z
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