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| Given a [[fully nonlinear integro-differential equation]] $Iu=0$, [[uniformly elliptic]] with respect to certain [[class of operators]], sometimes an interior $C^{1,\alpha}$ estimate holds for some $\alpha>0$ (typically very small). Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.
| | 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: |
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| '''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
| | \[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \] |
| Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds
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| \[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]
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| A theorem as above is known to hold under some assumptions on the [[nonlocal operator]] $I$. A list of valid assumptions is provided below.
| | 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 $E$. |
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| Note that the result is stated for general [[fully nonlinear integro-differential equations]], but the most important cases to apply it are the [[Isaacs equation]] and [[Bellman equation]].
| | 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. |
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| == Idea of the proof ==
| | 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 |
| The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities
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| \[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \]
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| where $M^\pm_{\mathcal L}$ are the [[extremal operators]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.
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| There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.
| | \[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \] |
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| The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.
| | == Definition == |
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| ==Examples for which the estimate holds ==
| | 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 whose [[Nonlocal mean curvature]] $H_s$ is identically zero (see next section), that is |
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| === Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels === | | \[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\] |
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| The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and [[uniformly elliptic]] with respect to the class of kernels satisfying the following hypothesis for some $\rho_0$ small enough<ref name="CS"/>.
| | In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. |
| \begin{align*}
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| \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
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| \int_{\R^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} \mathrm d y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}$)}
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| \end{align*}
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| === Variant if the kernel tails are $C^1$ ===
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| A small variation of the previous result is to assume the class of kernels satisfying the slightly stronger assumptions. A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is <ref name="CS2"/>
| | 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 |
| \begin{align*} | |
| \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\ | |
| \nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.}
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| \end{align*} | |
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| Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate
| | \[J_s(E) \leq J_s(F) \] |
| \[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \] | |
| Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]] <ref name="CS2"/>.
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| === A class of non-differentiable kernels ===
| | Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$. |
| A scale invariant class for which interior $C^{1,\alpha}$ regularity holds and the kernels can be very irregular is given by the following hypothesis<ref name="CS2"/>
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| \begin{align*}
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| K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{|y|^{n+s}} \\
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| \lambda &\leq a_1(y) \leq \Lambda \\
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| |a_2| &\leq \eta \\
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| |\nabla a_1(y)| &\leq \frac{C_1}{|y|} \qquad \text{in } \R^n \setminus \{0\}
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| \end{align*}
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| for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)
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| === Isaacs equation with variable coefficients but close to constant === | | <div style="background:#DDEEFF;"> |
| If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>. The family of integro-differential operators has kernels which are the sum of a fixed term $a_0$ (the same for all kernels in the class) and a small term which can depend on $x$.
| | <blockquote> |
| \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\]
| | '''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. |
| such that we have for $\eta$ small enough and any $\alpha$, $\beta$,
| | </blockquote> |
| \begin{align*}
| | </div> |
| |a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\
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| \lambda &\leq a_0(y) \leq \Lambda \\
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| |\nabla a_0(y)| &\leq C |y|^{-1}
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| \end{align*}
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| (note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)
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| === Isaacs equation with continuous coefficients ===
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| If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates <ref name="CS2"/>.
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| \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\]
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| such that for every $\alpha$, $\beta$ we have
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| \begin{align*}
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| \lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\
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| \nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\
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| |a_{\alpha \beta}(x_1,y) - a_{\alpha \beta}(x_2,y)| &\leq c(|x_1-x_2|) && \text{for some uniform modulus of continuity $c$}.
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| \end{align*}
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| | == Nonlocal mean curvature == |
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| | == Surfaces minimizing non-local energy functionals == |
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| == References == | | == The Caffarelli-Roquejoffre-Savin Regularity Theorem== |
| {{reflist|refs=
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| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
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| <ref name="CS2">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity results for nonlocal equations by approximation | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref>
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| }}
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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 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 whose Nonlocal mean curvature $H_s$ is identically zero (see next section), that is
\[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]
In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$.
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