Nonlocal minimal surfaces

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

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