Nonlocal minimal surfaces

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In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that are minimizers or critical points (within a class of given admissible configurations) of 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 get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials

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



Following the most accepted convention for minimal surfaces, a (classical) nonlocal minimal surface is (given $s\in (0,1)$) the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\Sigma$ is at least $C^{1,s+\epsilon}$ and more importantly,

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

In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. The quantity $H_s(x)$ is called the "Nonlocal mean curvature of order $s$ of $\Sigma$ at $x$", or briefly, "Nonlocal mean curvature".

Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have

\[J_s(E) \leq J_s(F) \]

Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$.

Note 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.

Nonlocal mean curvature

The scalar quantity

\[ H_s : \Sigma \to \mathbb{R} \] \[ H_s (x) := -C_{n,s} \int_{\mathbb{R}^n}\frac{\chi_E(y)-\chi_{E^c(y)}}{|x-y|^{n+s}}dy \]

Is called the nonlocal mean curvature of $\Sigma$ (or $E$) at $x$, its a real valued function defined on $\Sigma$. Like the usual mean curvature, it measures in some average sense the deviation of $\Sigma$ from its tangent hyperplane at $x$ (note that if $\Sigma$ is a hyperplane, then trivially $H_s \equiv 0$).

Mean curvature vs Nonlocal mean curvature: Whereas the standard mean curvature measures mean deviation from flatness at the infinitesimal scale, the nonlocal mean curvature takes into account all scales (infinitesimal and positive scales). Note further that for the mean curvature to be classically defined $\Sigma$ must be at least a $C^2$ surface, however, for $H_s$ one needs only a bit more than $C^{1,s}$ regularity.

As the kernel is invariant under Euclidean symmetries, we conclude for instance that any sphere $\partial B_r(x_0)$ has constant nonlocal mean curvature. Moreover, via a change of variables in the integral defining $H_s$ ($x \to x_0 \to rx$) one can see that a sphere of radius $r$ has mean curvature equal to $c_{n,s}r^{-s}$ (Note that $s=1$ gives the local mean curvature).

An alternative expression for the nonlocal mean curvature can be obtained via integration by parts (when $\Sigma$ is smooth enough). It is based on the identity

\[ \nabla_y \cdot \left ( \frac{y-x}{|y-x|^{n+s}} \right ) = \frac{n}{|x-y|^{n+s}}-(n+s) \frac{(y-x) \cdot (y-x) }{|x-y|^{n+s+2}} = -\frac{s}{|y-x|^{n+s}} \]

We use integration by parts (compute the integrals outside a ball $B_\epsilon(x)$ and then letting $\epsilon \to 0$). If $\nu$ denotes the outer normal to $\Sigma$, we get

\[ H_s(x) = -\frac{2C_{n,s}}{s}\int_{\Sigma} \frac{(x-y) \cdot \nu(y)}{|x-y|^{n+s}} d\sigma(y) \]

Surfaces minimizing non-local energy functionals

A variational formulation of the Plateau problem for the nonlocal energy functional $J_s$ is as follows:

"Given a bounded Lipschitz domain $\Omega$ and a (possibly unbounded) set $E_0$, minimize $J_s$ among all sets which agree with $E_0$ outside $\Omega$."

That there exists a unique minimizer is always true by the Sobolev embedding and the lower-semicontinuity of $J_s$ with respect to $L^1$ convergence. Of course to carry the argument one needs to assume that there is at least on set $E$ such that $\chi_E \in H^s$ and $E$ agrees with $E_0$ outside $\Omega$ (namely $E \Delta E_0 \subset \Omega$).

In contrast to the perimeter functional, which is local, the functional Js(E) has the remarkable feature of behaving like a quadratic form. In particular, one has the following identity

\[ J_s(E) = L(E,E^c) \]

where for any pair of measurable sets $A,B$ we define

\[ L(A,B) := C_{n,s}\int_A \int_B \frac{1}{|x-y|^{n+s}}dxdy = L(B,A) \]

Therefore, if we denote $A^- = E \setminus F, A^+= F\setminus E$ then

\[ J_s(F) - J_s(E) = L(F,F^c) - L(E,E^c ) \]

\[ = \left [ L(A^-,E \setminus A^-) -L(A^-,E^c) \right ]- \left [ L(A^+,E)-L(A^+,(E \cup A^+)^c) \right ] +2 L(A^-,A^+) \geq 0 \]

The Caffarelli-Roquejoffre-Savin Regularity Theorem

A regularity theory for solutions of the above variational problem has been developed recently [1]. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:

"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^*E \subset \partial E$ and $\partial E \setminus \partial^* E$ is a closed set of Hausdorff dimension at most $n-2$."

Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-8$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is not known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces. Even in two dimensions, the possibility of a singularity is not ruled out.


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