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| The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form
| | 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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| \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \] | | \[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \] |
| where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
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| \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \] | |
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| The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.
| | 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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| == Absolutely continuous measure ==
| | 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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| In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
| | 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 |
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| We keep this assumption in all the examples below.
| | \[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \] |
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| == Purely integro-differential operator == | | == Definition == |
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| In this case we neglect the local part of the operator
| | 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, |
| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
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| == Symmetric kernels ==
| | \[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\] |
| If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.
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| In the purely integro-differentiable case, it reads as | | 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". |
| \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \] | |
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| The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
| | 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 |
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| == Translation invariant operators ==
| | \[J_s(E) \leq J_s(F) \] |
| In this case, all coefficients are independent of $x$.
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| \[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \] | |
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| == The fractional Laplacian ==
| | Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$. |
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| The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
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| | <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> |
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| \[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
| | == Nonlocal mean curvature == |
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| == Stable operators == | | == Surfaces minimizing non-local energy functionals == |
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| These are the operators whose kernel is homogeneous in $y$
| | == The Caffarelli-Roquejoffre-Savin Regularity Theorem== |
| \[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
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| They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.
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| == Uniformly elliptic of order $s$ ==
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| This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
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| \[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
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| The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
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| An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.
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| == Smoothness class $k$ of order $s$ ==
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| This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
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| \[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
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| == Order strictly below one ==
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| If a non symmetric kernel $K$ satisfies the extra local integrability assumption
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| \[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
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| then the extra gradient term is not necessary in order to define the operator.
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
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| The modification in the integro-differential part of the operator becomes an extra drift term.
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| A uniformly elliptic operator of order $s<1$ satisfies this condition.
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| == Order strictly above one ==
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| If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
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| \[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
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| then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
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| The modification in the integro-differential part of the operator becomes an extra drift term.
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| A uniformly elliptic operator of order $s>1$ satisfies this condition.
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| == More singular/irregular kernels ==
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| The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />. The integro-differential operators have the form
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| \begin{align*}
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| \int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
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| \int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
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| \int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
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| \end{align*}
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| For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$,
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| * $K(x,h) \geq 0$ for all $h\in\R^d$.
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| This is a basic assumption for the integral operator to be a legit diffusion operator.
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| * For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
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| This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.
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| * For every $r>0$, there exists a set $A_r$ such that
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| ** $A_r \subset B_{2r} \setminus B_r$.
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| ** $A_r$ is symmetric in the sense that $A_r = -A_r$.
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| ** $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
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| ** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
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| Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|.
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| \]
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| This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.
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| * For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]
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| This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.
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| == Indexed by a matrix ==
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| In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
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| \[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
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| This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$. This property has been exploited in order to obtain an ABP estimate in integral form. <ref name="GS"/>
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| == Second order elliptic operators as limits of purely integro-differential ones ==
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| Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators
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| \[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
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| define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. We get
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| \[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\]
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| where $a_{ij}$ is determined by the quadratic form
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| \[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\]
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| A class of kernels that is big enough to recover all translation invariant elliptic operators of the form $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
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| \[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
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| where $A$ is an invertible symmetric matrix.
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| Conversely, the condition
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| \[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\]
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| defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.<ref name="ros2014regularity" /> If we let the operator depend on $x$, it is an outstanding [[Open problems#Hölder estimates for singular integro-differential equations|open problem]] whether this condition alone (for every value of $x$) ensures [[Holder estimates]] for the integro-differential equation.
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| == Characterization via global maximum principle ==
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| A bounded linear operator
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| \[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]
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| is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have
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| \[ (Lu)(x_0) \leq 0 \]
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| It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form
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| \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
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| where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying
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| \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
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| and $A(x),c(x)$ and $b(x)$ are bounded.
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| == See also ==
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| * [[Fractional Laplacian]]
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| * [[Levy processes]]
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| * [[Dirichlet form]]
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| == References ==
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| {{reflist|refs=
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| <ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</ref>
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| <ref name="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de $C_k^\infty(\mathbb{R}^n)$ dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</ref>
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| <ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
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| <ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
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| <ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</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 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
Surfaces minimizing non-local energy functionals
The Caffarelli-Roquejoffre-Savin Regularity Theorem