Nonlocal minimal surfaces

In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the energy functional:

\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]

It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.

Classically, minimal surfaces (or generally surfaces of constant mean curvature ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.

Nonlocal minimal surfaces then, describe physical phenomena where the interaction potential does not decay fast enough as particles are apart, so that two particles on different phases and far from the interface still contribute a non-trivial amount to the total interaction energy, in particular, one may consider much more general energy functionals corresponding to different interaction potentials

\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]

Definition

Following the most accepted convention for minimal surfaces, a nonlocal minimal surface is the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\chi_E \in \dot{H}^{s/2}$ and whose Nonlocal mean curvature is identically zero, that is

\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]

Note that for this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as the minimal surface, and no confusion should arise from this.