Nonlocal Evans-Krylov theorem and Nonlocal minimal surfaces: Difference between pages

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The classical [[Evans-Krylov theorem]] <ref name="E"/> <ref name="K"/> says that convex or concave fully nonlinear elliptic equations have $C^{2,\alpha}$ (therefore classical) solutions. This type of equations can be written as a Hamilton-Jacobi-Bellman equation.
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:
\[ \sup_\beta a_{ij}^\beta \partial_{ij} u = f \]
for a family of uniformly elliptic coefficients $a_{ij}^\alpha$.


A purely integro-differential version of this theorem<ref name="CS3"/> says that solutions of an integro-differential [[Bellman equation]] of the form
\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
\[ \sup_\beta \int_{\R^n} (u(x+y) - u(x)) K_\beta (y) \mathrm d y = 0 \qquad \text{in } B_1\]
are $C^{s+\alpha}(B_{1/2})$ (which implies that they are classical) if the kernels satisfy the following assumptions
\begin{align*}
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{uniform ellipticity of order $s$} \\
D^2 K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s+2}} && \text{Decay of the tails of $K$ in $C^2$} \\
K(y) &= K(-y) && \text{symmetry}
\end{align*}


The $C^{s+\alpha}$ estimate '''does not blow up as $s \to 2$'''. Thus, the result is a true generalization of Evans-Krylov theorem.
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 $E$.  


Note that the result is relevant only if $s>1$, otherwise it is a weaker result compared to the [[differentiability estimates|$C^{1,\alpha}$ estimates]].  
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.


The hypothesis above are most probably not optimal. But unlike the [[differentiability estimates|$C^{1,\alpha}$ estimates]], no extension of this result has been done.
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


== References ==
\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
{{reflist|refs=
 
<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
== Definition ==  
<ref name="E">{{Citation | last1=Evans | first1=Lawrence C. | title=Classical solutions of fully nonlinear, convex, second-order elliptic equations | url=http://dx.doi.org/10.1002/cpa.3160350303 | doi=10.1002/cpa.3160350303 | year=1982 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=35 | issue=3 | pages=333–363}}</ref>
 
<ref name="K">{{Citation | last1=Krylov | first1=N. V. | title=Boundedly inhomogeneous elliptic and parabolic equations | year=1982 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=46 | issue=3 | pages=487–523}}</ref>
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 $\Sigma$ of order $s$ at $x$".
 
 
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 $\Omega$.
 
<div style="background:#DDEEFF;">
<blockquote>
'''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.  
</blockquote>
</div>
 
 
== Nonlocal mean curvature ==  
 
== Surfaces minimizing non-local energy functionals ==
 
== The Caffarelli-Roquejoffre-Savin Regularity Theorem==

Revision as of 13:28, 31 May 2011

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 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 (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 $\Sigma$ of order $s$ at $x$".


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

Surfaces minimizing non-local energy functionals

The Caffarelli-Roquejoffre-Savin Regularity Theorem