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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 | | The extremal operator associated to some class of linear operators $\mathcal{L}$ represent the maximal and minimal value that $Lu(x)$ can take from all possible choices of $L \in \mathcal L$. |
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| \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
| | The extremal operators are used to define [[uniformly elliptic|uniform ellipticity]] for nonlocal operators. In fact, the extremal operators are also the maximal and minimal nonlinear uniformly elliptic operators with respect to $\mathcal L$ that vanish at zero. |
| 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.
| | Given any family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: |
| | | \begin{align*} |
| == Absolutely continuous measure ==
| | M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\ |
| | | M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x) |
| In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
| | \end{align*} |
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| We keep this assumption in all the examples below.
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| == Purely integro-differential operator ==
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| In this case we neglect the local part of the operator
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
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| == Symmetric kernels ==
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| 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
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| \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]
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| The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
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| == Translation invariant operators ==
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| 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 ==
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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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| \[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
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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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| == 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 $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{s-1} u \right] (x)$ for some coefficients $a_{ij}$.
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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. 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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| == 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 (see <ref name="C65"/> and <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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| | If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula |
| | \begin{align*} |
| | P^+(D^2 u) &= \Lambda \ \mathrm{tr}(D^2u^+) - \lambda \ \mathrm{tr}(D^2u^-)\\ |
| | P^-(D^2 u) &= \lambda \ \mathrm{tr}(D^2u^+) - \Lambda \ \mathrm{tr}(D^2u^-) |
| | \end{align*} |
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| | If $\mathcal{L}$ consists of all [[linear integro-differential operator|symmetric purely integro-differential operators, uniformly elliptic of order $s$]], then the extremal operators have the formula<ref name="S"/> |
| | \begin{align*} |
| | M^+\, u &= \int_{\R^n} \left( \Lambda \delta u(x,y)^+ - \lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \\ |
| | M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y |
| | \end{align*} |
| | where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the ''monster'' Pucci operators". |
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| == References == | | == References == |
| {{reflist|refs= | | {{reflist|refs= |
| <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> | | <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref> |
| <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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| }} | | }} |
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| | [[Category:Fully nonlinear equations]] |