Comparison principle - Revision history

imported>Luis: /* Proper elliptic equations with $x$ dependence */

2011-07-31T18:55:02Z

Proper elliptic equations with $x$ dependence

← Older revision		Revision as of 13:55, 31 July 2011
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	The comparison principle holds provided that the following assumptions are satisfied:		The comparison principle holds provided that the following assumptions are satisfied:
	* All integro-differential operators $I_\alpha$ are in the [[Levy-Ito form]]		* All integro-differential operators $I_\alpha$ are in the [[Levy-Ito form]]
	\[ I_\alpha = \int_{\R^n} (u(x+j_\alpha(x,z)) - u(x) - \nabla u(x) \cdot j_\alpha(x,z) \mathbb{1}_B(z)) \mu_\alpha(dz). \]		\[ I_\alpha u (x)= \int_{\R^n} (u(x+j_\alpha(x,z)) - u(x) - \nabla u(x) \cdot j_\alpha(x,z) \mathbb{1}_B(z)) \mu_\alpha(dz). \]
	* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.		* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.
	\begin{align*}		\begin{align*}
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	</blockquote>		</blockquote>
	</div>		</div>


	=== Uniformly elliptic equations without $x$ dependence ===		=== Uniformly elliptic equations without $x$ dependence ===

imported>Luis at 20:07, 15 June 2011

2011-06-15T20:07:19Z

← Older revision		Revision as of 15:07, 15 June 2011
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	</blockquote>		</blockquote>

	This assumption on the family $\mathcal L$ is ~~easy to satisfy~~. The strong assumption is translation invariance. The proof of this comparison principle uses inf and sup-convolutions.		This assumption on the family $\mathcal L$ is a very mild form of uniform ellipticity. The strong assumption is translation invariance. The proof of this comparison principle uses inf and sup-convolutions.

imported>Luis at 20:06, 15 June 2011

2011-06-15T20:06:37Z

← Older revision		Revision as of 15:06, 15 June 2011
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	=== Proper elliptic equations with $x$ dependence ===		=== Proper elliptic equations with $x$ dependence ===
	A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form		A fairly general comparison principle <ref name="BI"/> <ref name="BIC2"/> is available for [[fully nonlinear integro-differential equations]] of the form
	\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]		\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]
	where $\{I_\alpha u\}_{\alpha \in A}$ is a family of [[linear integro-differential operators]].		where $\{I_\alpha u\}_{\alpha \in A}$ is a family of [[linear integro-differential operators]].
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	</blockquote>		</blockquote>

	This assumption on the family $\mathcal L$ is easy to satisfy. The strong assumption is translation invariance. The proof of this comparison principle ~~is based on~~ inf and sup-convolutions.		This assumption on the family $\mathcal L$ is easy to satisfy. The strong assumption is translation invariance. The proof of this comparison principle uses inf and sup-convolutions.


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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>		<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>
	<ref name="BI">{{Citation \| last1=Barles \| first1=Guy \| last2=Imbert \| first2=Cyril \| title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited \| url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 \| doi=10.1016/j.anihpc.2007.02.007 \| year=2008 \| journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire \| issn=0294-1449 \| volume=25 \| issue=3 \| pages=567–585}}</ref>		<ref name="BI">{{Citation \| last1=Barles \| first1=Guy \| last2=Imbert \| first2=Cyril \| title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited \| url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 \| doi=10.1016/j.anihpc.2007.02.007 \| year=2008 \| journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire \| issn=0294-1449 \| volume=25 \| issue=3 \| pages=567–585}}</ref>
			<ref name="BIC2">{{Citation \| last1=Barles \| first1=G. \| last2=Chasseigne \| first2=Emmanuel \| last3=Imbert \| first3=Cyril \| title=On the Dirichlet problem for second-order elliptic integro-differential equations \| url=http://dx.doi.org/10.1512/iumj.2008.57.3315 \| doi=10.1512/iumj.2008.57.3315 \| year=2008 \| journal=Indiana University Mathematics Journal \| issn=0022-2518 \| volume=57 \| issue=1 \| pages=213–246}}</ref>
	}}		}}

imported>Luis at 04:12, 14 June 2011

2011-06-14T04:12:42Z

← Older revision		Revision as of 23:12, 13 June 2011
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	== Assumptions for which the comparison principle holds ==		== Assumptions for which the comparison principle holds ==

	=== ~~General~~ elliptic equations with $x$ dependence ===		=== Proper elliptic equations with $x$ dependence ===
	A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form		A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form
	\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]		\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]

imported>Luis at 03:47, 14 June 2011

2011-06-14T03:47:32Z

← Older revision		Revision as of 22:47, 13 June 2011
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	* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.		* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.
	\begin{align*}		\begin{align*}
	\int_B \|j(x,z)\|^s \mu(dz) &\leq C, \qquad \int_{\R^n \setminus B} \mu(dz) \leq C,\\		\int_B \|j(x,z)\|^2 \mu(dz) &\leq C, \qquad \int_{\R^n \setminus B} \mu(dz) \leq C,\\
	\int_{\R^n} \|j(x,z)-j(y,z)\|^2 \mu(dz) &\leq C\|x-y\|^2, \text{ and } \int_{\R^n \setminus B} \|j(x,z)-j(y,z)\|^2 \mu(dz) \leq C\|x-y\|.		\int_{\R^n} \|j(x,z)-j(y,z)\|^2 \mu(dz) &\leq C\|x-y\|^2, \text{ and } \int_{\R^n \setminus B} \|j(x,z)-j(y,z)\|^2 \mu(dz) \leq C\|x-y\|.
	\end{align*}		\end{align*}
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	\[ F(x,u,p,X,\{i_\alpha\}) - F(x,v,p,X,\{i_\alpha\}) \geq \gamma (v-u).\]		\[ F(x,u,p,X,\{i_\alpha\}) - F(x,v,p,X,\{i_\alpha\}) \geq \gamma (v-u).\]
	* One of the following two holds		* One of the following two holds
	*# For any $R >0$, there exist moduli of continuity $\omega$ and $\omega_R$ such that, for any $\|x\|,\|y\|<R$, $\|v\|<R$, $\{i_\alpha\} \in \R^A$, and for any two symmetric matrices $X$ and $Y$ satisfying \[ \left[ \begin{matrix} X & 0 \\ 0 & -Y \end{matrix} \right] \leq \frac 1 \varepsilon \left[ \begin{matrix} I & -I \\ -I & I \end{matrix} \right] +r(\beta) \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right]\] for some $\varepsilon>0$ and $r(\beta)\to 0$ as $\beta\to 0$, then, if $s(\beta)\to 0$ as $\beta \to 0$, we have \~~begin{equation} \label{e1}~~ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y)+s(\beta),X,\{i_\alpha\})\geq\omega(\beta) +\omega_R(\|x-y\|+\varepsilon^{-1}\|x-y\|^2) \~~end{equation}~~		*# For any $R >0$, there exist moduli of continuity $\omega$ and $\omega_R$ such that, for any $\|x\|,\|y\|<R$, $\|v\|<R$, $\{i_\alpha\} \in \R^A$, and for any two symmetric matrices $X$ and $Y$ satisfying \begin{equation} \label{e1} \left[ \begin{matrix} X & 0 \\ 0 & -Y \end{matrix} \right] \leq \frac 1 \varepsilon \left[ \begin{matrix} I & -I \\ -I & I \end{matrix} \right] +r(\beta) \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right] \end{equation} for some $\varepsilon>0$ and $r(\beta)\to 0$ as $\beta\to 0$, then, if $s(\beta)\to 0$ as $\beta \to 0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y)+s(\beta),X,\{i_\alpha\})\geq\omega(\beta) +\omega_R(\|x-y\|+\varepsilon^{-1}\|x-y\|^2)\]
	*# For any $R>0$, $F$ is uniformly continuous on $R^n\times [−R,R] \times B_R \times D_R \times R^A$ where $D_R := \{X \in \R^{n \times n}, \text{symmetric}, \|X\|\leq R\}$ and there exist a modulus of continuity $\omega_R$ such that, for any $x,y\in \R^n$ , $\|v\|\leq R$, $\{i_\alpha\}\in \R^A$ and for any $X,Y \in \R^{n\times n}$ symmetric satisfying \eqref{e1} and $\varepsilon >0$, we have		*# For any $R>0$, $F$ is uniformly continuous on $R^n\times [−R,R] \times B_R \times D_R \times R^A$ where $D_R := \{X \in \R^{n \times n}, \text{symmetric}, \|X\|\leq R\}$ and there exist a modulus of continuity $\omega_R$ such that, for any $x,y\in \R^n$ , $\|v\|\leq R$, $\{i_\alpha\}\in \R^A$ and for any $X,Y \in \R^{n\times n}$ symmetric satisfying \eqref{e1} and $\varepsilon >0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y),X,\{i_\alpha\})\geq \omega_R(\varepsilon^{-1}\|x-y\|^2+\|x-y\|+r(\beta))\]
	* $F(x,u,p,X,\{i_\alpha\})$ is Lipschitz continuous in $\{i_\alpha\}$ uniformly respect to all other variables.		* $F(x,u,p,X,\{i_\alpha\})$ is Lipschitz continuous in $\{i_\alpha\}$ uniformly respect to all other variables.

imported>Luis at 17:30, 12 June 2011

2011-06-12T17:30:55Z

← Older revision		Revision as of 12:30, 12 June 2011
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	A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form		A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form
	\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]		\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]
	where $\{I_\alpha u\}$ is a family of [[linear integro-differential operators]].		where $\{I_\alpha u\}_{\alpha \in A}$ is a family of [[linear integro-differential operators]].

	The comparison principle holds provided that the following assumptions are satisfied:		The comparison principle holds provided that the following assumptions are satisfied:
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	<div style="background:#DDEEFF;">		<div style="background:#DDEEFF;">
	<blockquote>		<blockquote>
	'''Note'''. Any linear integro-differential operator which is written in Levy-Ito form can be written in the usual form (with $j(x,z)=z$ and an $x$ dependent jump measure $\mu(x,dz)$) by a change of variables. Conversely, any integro-differential operator with an $x$ dependent $\mu$ can be written in Levy-Ito form for several choices of $j$, as long as $\int_{\R^n} \mu(x,dz) = +\infty$ for all $x$. The reason for assuming that all operators $I_\alpha$ are in the Levy-Ito form is to state the right continuity condition in the second assumption. A comparison principle for a family of integro-differential operators $I_\alpha$ with measures $\mu(x,dz)$ (or kernels $K(x,y)$) which are continuous with respect to $x$, but without any assumption on their Levy-Ito form, is currently an open problem.		'''Note'''. Any linear integro-differential operator which is written in Levy-Ito form can be written in the usual form (with $j(x,z)=z$ and an $x$ dependent jump measure $\mu(x,dz)$) by a change of variables. Conversely, any integro-differential operator with an $x$ dependent $\mu$ can be written in Levy-Ito form for several choices of $j$, as long as $\int_{\R^n} \mu(x,dz) = +\infty$ for all $x$. The reason for assuming that all operators $I_\alpha$ are in the Levy-Ito form is to state the right continuity condition in the second assumption. A comparison principle for a family of integro-differential operators $I_\alpha$ with measures $\mu(x,dz)$ (or kernels $K(x,y)$) which are continuous with respect to $x$, but without any continuity assumption with respect to their Levy-Ito form, is currently an open problem.
	</blockquote>		</blockquote>
	</div>		</div>

imported>Luis at 17:24, 12 June 2011

2011-06-12T17:24:21Z

← Older revision		Revision as of 12:24, 12 June 2011
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	== Assumptions for which the comparison principle holds ==		== Assumptions for which the comparison principle holds ==

			=== General elliptic equations with $x$ dependence ===
			A fairly general comparison principle <ref name="BI"/> is available for [[fully nonlinear integro-differential equations]] of the form
			\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]
			where $\{I_\alpha u\}$ is a family of [[linear integro-differential operators]].

			The comparison principle holds provided that the following assumptions are satisfied:
			* All integro-differential operators $I_\alpha$ are in the [[Levy-Ito form]]
			\[ I_\alpha = \int_{\R^n} (u(x+j_\alpha(x,z)) - u(x) - \nabla u(x) \cdot j_\alpha(x,z) \mathbb{1}_B(z)) \mu_\alpha(dz). \]
			* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.
			\begin{align*}
			\int_B \|j(x,z)\|^s \mu(dz) &\leq C, \qquad \int_{\R^n \setminus B} \mu(dz) \leq C,\\
			\int_{\R^n} \|j(x,z)-j(y,z)\|^2 \mu(dz) &\leq C\|x-y\|^2, \text{ and } \int_{\R^n \setminus B} \|j(x,z)-j(y,z)\|^2 \mu(dz) \leq C\|x-y\|.
			\end{align*}
			* There exist a $\gamma>0$ such that for all $x \in \R^n$, $u,v \in \R$, $p \in \R^n$, $X$ a symmetric matrix in $\R^{n \times n}$ and $\{i_\alpha\} \in \R^A$,
			\[ F(x,u,p,X,\{i_\alpha\}) - F(x,v,p,X,\{i_\alpha\}) \geq \gamma (v-u).\]
			* One of the following two holds
			*# For any $R >0$, there exist moduli of continuity $\omega$ and $\omega_R$ such that, for any $\|x\|,\|y\|<R$, $\|v\|<R$, $\{i_\alpha\} \in \R^A$, and for any two symmetric matrices $X$ and $Y$ satisfying \[ \left[ \begin{matrix} X & 0 \\ 0 & -Y \end{matrix} \right] \leq \frac 1 \varepsilon \left[ \begin{matrix} I & -I \\ -I & I \end{matrix} \right] +r(\beta) \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right]\] for some $\varepsilon>0$ and $r(\beta)\to 0$ as $\beta\to 0$, then, if $s(\beta)\to 0$ as $\beta \to 0$, we have \begin{equation} \label{e1} F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y)+s(\beta),X,\{i_\alpha\})\geq\omega(\beta) +\omega_R(\|x-y\|+\varepsilon^{-1}\|x-y\|^2) \end{equation}
			*# For any $R>0$, $F$ is uniformly continuous on $R^n\times [−R,R] \times B_R \times D_R \times R^A$ where $D_R := \{X \in \R^{n \times n}, \text{symmetric}, \|X\|\leq R\}$ and there exist a modulus of continuity $\omega_R$ such that, for any $x,y\in \R^n$ , $\|v\|\leq R$, $\{i_\alpha\}\in \R^A$ and for any $X,Y \in \R^{n\times n}$ symmetric satisfying \eqref{e1} and $\varepsilon >0$, we have
			* $F(x,u,p,X,\{i_\alpha\})$ is Lipschitz continuous in $\{i_\alpha\}$ uniformly respect to all other variables.

			Note that the assumptions are designed so that Ishii's lemma can be reproduced in this setting.

			<div style="background:#DDEEFF;">
			<blockquote>
			'''Note'''. Any linear integro-differential operator which is written in Levy-Ito form can be written in the usual form (with $j(x,z)=z$ and an $x$ dependent jump measure $\mu(x,dz)$) by a change of variables. Conversely, any integro-differential operator with an $x$ dependent $\mu$ can be written in Levy-Ito form for several choices of $j$, as long as $\int_{\R^n} \mu(x,dz) = +\infty$ for all $x$. The reason for assuming that all operators $I_\alpha$ are in the Levy-Ito form is to state the right continuity condition in the second assumption. A comparison principle for a family of integro-differential operators $I_\alpha$ with measures $\mu(x,dz)$ (or kernels $K(x,y)$) which are continuous with respect to $x$, but without any assumption on their Levy-Ito form, is currently an open problem.
			</blockquote>
			</div>


			=== Uniformly elliptic equations without $x$ dependence ===
			A simple comparison principle can be shown for [[fully nonlinear integro-differential equations]] which are translation invariant and uniformly elliptic respect to some family $\mathcal L$ of linear operators satisfying some assumptions <ref name="CS"/>. In fact, if $Iu \geq 0$ and $Iv \leq 0$ in some domain $\Omega$, then $M_{\mathcal L}^+ (u-v) \geq 0$ where $M_{\mathcal L}$ is the [[extremal operator]] respect to the class $\mathcal L$.

			The family $\mathcal L$ is assumed to be a family of purely integro-differential [[linear integro-differential operators]] which are translation invariant and satisfy the following assumption:

			<blockquote> There is a constant $R_0\geq 1$ so that for every $R>R_0$, there exists
			a $\delta>0$ (which could depend on $R$) such that for any operator $L$ in $\mathcal L$, we have that
			$L \varphi > \delta$ in $B_R$, where $\varphi$ is given by $\varphi(x)=\min(1,\|x\|^2/R^3)$.
			</blockquote>

			This assumption on the family $\mathcal L$ is easy to satisfy. The strong assumption is translation invariance. The proof of this comparison principle is based on inf and sup-convolutions.


			== References ==
			{{reflist\|refs=
			<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>
			<ref name="BI">{{Citation \| last1=Barles \| first1=Guy \| last2=Imbert \| first2=Cyril \| title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited \| url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 \| doi=10.1016/j.anihpc.2007.02.007 \| year=2008 \| journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire \| issn=0294-1449 \| volume=25 \| issue=3 \| pages=567–585}}</ref>
			}}

imported>Luis at 16:17, 12 June 2011

2011-06-12T16:17:55Z

imported>Luis: Created page with "The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great..."

2011-06-12T15:18:49Z

Created page with "The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great..."

New page

The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions.

The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the [[maximum principle]]. The uniqueness of the solution of the equation is an immediate consequence.

== General statement ==

The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions.

Other boundary conditions require appropriate modifications.

=== Elliptic case ===
We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that
$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in an open domain $\Omega$, and
$u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.

=== Parabolic case ===
We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that
$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $(0,T] \times \Omega$, and
$u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.

← Older revision		Revision as of 11:17, 12 June 2011
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	== General statement ==		== General statement ==

	The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions.		The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions. The main difference with the local case, is that for nonlocal equations the Dirichlet condition has to be taken in the whole complement of the domain $\Omega$ instead of only the boundary.

	Other boundary conditions require appropriate modifications.		Other boundary conditions require appropriate modifications.
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	We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.		We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.

	Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that		Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $\overline \Omega$ respectively, where $\Omega$ is an open domain,
	$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in ~~an open domain~~ $\Omega$, and		$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $\Omega$, and
	$u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.		$u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.

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	We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.		We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.

	Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that		Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $[0,T] \times \overline \Omega$ respectively,
	$Iu \~~geq~~ 0$ and $Iv \~~leq~~ 0$ in the viscosity sense in $(0,T] \times \Omega$, and		$Iu \leq 0$ and $Iv \geq 0$ in the viscosity sense in $(0,T] \times \Omega$, and
	$u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.		$u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.

			== Assumptions for which the comparison principle holds ==