Linear integro-differential operator - Revision history

imported>Luis: /* Symmetric kernels */

2016-11-17T20:21:17Z

Symmetric kernels

← Older revision		Revision as of 15:21, 17 November 2016
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	\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \]		\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \]

	The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.		The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x-y)-2u(x))$.

	== Translation invariant operators ==		== Translation invariant operators ==

imported>Luis: /* Symmetric kernels */

2016-11-17T20:21:04Z

Symmetric kernels

← Older revision		Revision as of 15:21, 17 November 2016
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	In the purely integro-differentiable case, it reads as		In the purely integro-differentiable case, it reads as
	\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]		\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \]

	The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.		The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.

imported>Luis: /* References */

2016-01-26T17:50:58Z

References

← Older revision		Revision as of 12:50, 26 January 2016
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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>		<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>
	<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>		<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>
			<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>
	}}		}}

imported>Luis: /* Indexed by a matrix */

2016-01-26T17:50:24Z

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 $s$:		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 $s$:
	\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{\|y\|^{n+2+s}}. \]		\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{\|y\|^{n+2+s}}. \]
	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 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"/>

	== Second order elliptic operators as limits of purely integro-differential ones ==		== Second order elliptic operators as limits of purely integro-differential ones ==

imported>Luis: /* References */

2016-01-26T17:35:45Z

References

← Older revision		Revision as of 12:35, 26 January 2016
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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>		<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>
	<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>		<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>
			<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>
	}}		}}

imported>Luis: /* Second order elliptic operators as limits of purely integro-differential ones */

2016-01-26T17:35:22Z

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		Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators

	\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{\|y\|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]		\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{\|y\|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
			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
			\[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\]
			where $a_{ij}$ is determined by the quadratic form
			\[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\]

	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		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

	\[ K_A(y) = (2-\sigma) \frac{1}{\|Ay\|^{n+\sigma}},\]		\[ K_A(y) = (2-\sigma) \frac{1}{\|Ay\|^{n+\sigma}},\]
			where $A$ is an invertible symmetric matrix.

	~~where~~ $A$ is an ~~invertible symmetric matrix~~.		Conversely, the condition
			\[ \lambda \|e\|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda \|e\|^2, \qquad \text{for all vectors } e \in \R^d,\]
			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.

	== Characterization via global maximum principle ==		== Characterization via global maximum principle ==

imported>Luis: /* More singular/irregular kernels */

2016-01-26T17:21:21Z

More singular/irregular kernels

imported>Luis: /* More singular/irregular kernels */

2016-01-26T17:18:20Z

More singular/irregular kernels

← Older revision		Revision as of 12:18, 26 January 2016
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	== More singular/irregular kernels ==		== More singular/irregular kernels ==

	The concept of uniform ellipticity can be relaxed in various ways. The following family of ~~kernels~~ was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />		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
			\begin{align*}
			\int_{\R^d} \big(u(x+y) - u(x) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
			\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
			\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
			\end{align*}

	For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \R^d \to \R$ is assumed to satisfy the following assumptions.		For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \R^d \to \R$ is assumed to satisfy the following assumptions.

imported>Xavi: /* Stable operators */

2015-05-15T19:34:02Z

Stable operators

← Older revision		Revision as of 14:34, 15 May 2015
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	These are the operators whose kernel is homogeneous in $y$		These are the operators whose kernel is homogeneous in $y$
	\[ K(x,y)=\frac{a(x,y/\|y\|)}{\|y\|^{n+s}}.\]		\[ K(y)=\frac{a(y/\|y\|)}{\|y\|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/\|y\|)}{\|y\|^{n+s}}.\]
	They are the generators of stable Lévy processes.		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.

	== Uniformly elliptic of order $s$ ==		== Uniformly elliptic of order $s$ ==

imported>Luis: /* More singular/irregular kernels */

2015-04-08T20:15:26Z

More singular/irregular kernels

← Older revision		Revision as of 15:15, 8 April 2015
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	The concept of uniform ellipticity can be relaxed in various ways. The following family of kernels was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />		The concept of uniform ellipticity can be relaxed in various ways. The following family of kernels was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />

			For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \R^d \to \R$ is assumed to satisfy the following assumptions.
			* $K(h) \geq 0$ for all $h\in\R^d$.
			This is a basic assumption for the integral operator to be a legit diffusion operator.

			* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
			This assumption is more general than $K(y) \leq (2-\alpha) \Lambda \|y\|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.

			* For every $r>0$, there exists a set $A_r$ such that
			** $A_r \subset B_{2r} \setminus B_r$.
			** $A_r$ is symmetric in the sense that $A_r = -A_r$.
			** $\|A_r\| \geq \mu \|B_{2r} \setminus B_r\|$.
			** $K(h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
			Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu \|B_{2r} \setminus B_r\|.
			\]

			This assumption says that the lower bound $K(y) \geq (2-\alpha) \lambda \|y\|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.

			* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda \|1-\alpha\| r^{1-\alpha}. \]

			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 $\alpha \to 1$.

	== Indexed by a matrix ==		== Indexed by a matrix ==

← Older revision		Revision as of 12:21, 26 January 2016
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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		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
	\begin{align*}		\begin{align*}
	\int_{\R^d} \big(u(x+y) - u(x) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\		\int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
	\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\		\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, \\
	\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(h) \mathrm{d} h \qquad \text{ if } \alpha > 1.		\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.
	\end{align*}		\end{align*}

	For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \R^d \to \R$ is assumed to satisfy the following assumptions.		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 $x \in \Omega$,
	* $K(h) \geq 0$ for all $h\in\R^d$.		* $K(x,h) \geq 0$ for all $h\in\R^d$.
	This is a basic assumption for the integral operator to be a legit diffusion operator.		This is a basic assumption for the integral operator to be a legit diffusion operator.

	* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]		* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
	This assumption is more general than $K(y) \leq (2-\alpha) \Lambda \|y\|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.		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.

	* For every $r>0$, there exists a set $A_r$ such that		* For every $r>0$, there exists a set $A_r$ such that
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	** $A_r$ is symmetric in the sense that $A_r = -A_r$.		** $A_r$ is symmetric in the sense that $A_r = -A_r$.
	** $\|A_r\| \geq \mu \|B_{2r} \setminus B_r\|$.		** $\|A_r\| \geq \mu \|B_{2r} \setminus B_r\|$.
	** $K(h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.		** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
	Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu \|B_{2r} \setminus B_r\|.		Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu \|B_{2r} \setminus B_r\|.
	\]		\]

	This assumption says that the lower bound $K(y) \geq (2-\alpha) \lambda \|y\|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.		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.

	* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda \|1-\alpha\| r^{1-\alpha}. \]		* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda \|1-\alpha\| r^{1-\alpha}. \]