Harnack inequality - Revision history

imported>Luis: /* Concrete examples */

2015-04-23T15:57:03Z

Concrete examples

← Older revision		Revision as of 10:57, 23 April 2015
Line 47:		Line 47:
	It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.		It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.

	* An equation where the Harnack inequality fails.		* '''An equation where the Harnack inequality fails.'''

	There exist integro-differential equations of the form		There exist integro-differential equations of the form

imported>Luis at 15:56, 23 April 2015

2015-04-23T15:56:29Z

@@ Line 46: / Line 46: @@
 It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.
 == References ==
@@ Line 56: / Line 68: @@
 <ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
 <ref name="chang2014h">{{Citation | last1=Chang-Lara | first1= Hector | last2=Davila | first2= Gonzalo | title=Holder estimates for non-local parabolic equations with critical drift | journal=arXiv preprint arXiv:1408.0676}}</ref>
 }}

imported>Luis: /* Concrete examples */

2015-04-23T15:43:10Z

Concrete examples

← Older revision		Revision as of 10:43, 23 April 2015
Line 32:		Line 32:
	with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.		with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.

	In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="chang2014h"/>. It is a generalization of [[Krylov-Safonov]] theorem.		In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\\|f\\|$ refers to $\\|f\\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="chang2014h"/>. It is a generalization of [[Krylov-Safonov]] theorem.

	* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form		* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form

imported>Luis: /* Concrete examples */

2015-04-23T15:42:14Z

Concrete examples

← Older revision		Revision as of 10:42, 23 April 2015
Line 32:		Line 32:
	with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.		with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.

	In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="~~lara2011regularity~~"/>. It is a generalization of [[Krylov-Safonov]] theorem.		In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="chang2014h"/>. It is a generalization of [[Krylov-Safonov]] theorem.

	* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form		* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form

imported>Luis: /* References */

2015-04-23T15:41:34Z

References

← Older revision		Revision as of 10:41, 23 April 2015
Line 55:		Line 55:
	<ref name="BBCK">{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}</ref>		<ref name="BBCK">{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}</ref>
	<ref name="K">{{Citation \| last1=Kassmann \| first1=Moritz \| title=A priori estimates for integro-differential operators with measurable kernels \| url=http://dx.doi.org/10.1007/s00526-008-0173-6 \| doi=10.1007/s00526-008-0173-6 \| year=2009 \| journal=Calculus of Variations and Partial Differential Equations \| issn=0944-2669 \| volume=34 \| issue=1 \| pages=1–21}}</ref>		<ref name="K">{{Citation \| last1=Kassmann \| first1=Moritz \| title=A priori estimates for integro-differential operators with measurable kernels \| url=http://dx.doi.org/10.1007/s00526-008-0173-6 \| doi=10.1007/s00526-008-0173-6 \| year=2009 \| journal=Calculus of Variations and Partial Differential Equations \| issn=0944-2669 \| volume=34 \| issue=1 \| pages=1–21}}</ref>
	<ref name="~~lara2011regularity~~">{{Citation \| last1=Lara \| first1= ~~Héctor Chang~~ \| last2=~~Dávila~~ \| first2= Gonzalo \| title=~~Regularity~~ for ~~solutions of~~ non local parabolic equations \| journal=~~Calculus of Variations and Partial Differential Equations \| year=2011 \| pages=1--34~~}}</ref>		<ref name="chang2014h">{{Citation \| last1=Chang-Lara \| first1= Hector \| last2=Davila \| first2= Gonzalo \| title=Holder estimates for non-local parabolic equations with critical drift \| journal=arXiv preprint arXiv:1408.0676}}</ref>
	}}		}}

imported>Luis: /* Parabolic case */

2015-04-08T22:05:43Z

Parabolic case

← Older revision		Revision as of 17:05, 8 April 2015
Line 20:		Line 20:

	In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then		In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then
	\[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + \|\|f\|\| \right). \]		\[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + \\|f\\| \right). \]

	The norm $\|\|f\|\|$ may depend on the type of equation.		The norm $\\|f\\|$ may depend on the type of equation.

	== Concrete examples ==		== Concrete examples ==

imported>Luis: /* Elliptic case */

2015-04-08T22:05:24Z

Elliptic case

← Older revision		Revision as of 17:05, 8 April 2015
Line 13:		Line 13:

	In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then		In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then
	\[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|\|f\|\| \right). \]		\[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \\|f\\| \right). \]

	The norm $\|\|f\|\|$ may depend on the type of equation.		The norm $\\|f\\|$ may depend on the type of equation.

	== Parabolic case ==		== Parabolic case ==

imported>Luis: /* Concrete examples */

2013-06-13T18:46:00Z

Concrete examples

← Older revision		Revision as of 13:46, 13 June 2013
Line 32:		Line 32:
	with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.		with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.

	In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity">. It is a generalization of [[Krylov-Safonov]] theorem.		In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity"/>. It is a generalization of [[Krylov-Safonov]] theorem.

	* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form		* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form

imported>Luis: /* Concrete examples */

2013-06-13T18:45:49Z

Concrete examples

← Older revision		Revision as of 13:45, 13 June 2013
Line 28:		Line 28:
	The Harnack inequality as above is known to hold in the following situations.		The Harnack inequality as above is known to hold in the following situations.

	* '''Generalizad elliptic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form		* '''Generalizad elliptic and parabolic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form
	\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]		\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]
	with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.		with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda \|y\|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda \|y\|^{-n-s}$.

	In this case the elliptic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/>. It is a generalization of [[Krylov-Safonov]] theorem~~. The corresponding parabolic Harnack inequality with a uniform constant $C$ is not known~~.		In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $\|\|f\|\|$ refers to $\|\|f\|\|_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity">. It is a generalization of [[Krylov-Safonov]] theorem.

	* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form		* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form

imported>Luis: /* References */

2013-06-13T18:44:17Z

References

← Older revision		Revision as of 13:44, 13 June 2013
Line 55:		Line 55:
	<ref name="BBCK">{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}</ref>		<ref name="BBCK">{{Citation \| last1=Barlow \| first1=Martin T. \| last2=Bass \| first2=Richard F. \| last3=Chen \| first3=Zhen-Qing \| last4=Kassmann \| first4=Moritz \| title=Non-local Dirichlet forms and symmetric jump processes \| url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 \| doi=10.1090/S0002-9947-08-04544-3 \| year=2009 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=361 \| issue=4 \| pages=1963–1999}}</ref>
	<ref name="K">{{Citation \| last1=Kassmann \| first1=Moritz \| title=A priori estimates for integro-differential operators with measurable kernels \| url=http://dx.doi.org/10.1007/s00526-008-0173-6 \| doi=10.1007/s00526-008-0173-6 \| year=2009 \| journal=Calculus of Variations and Partial Differential Equations \| issn=0944-2669 \| volume=34 \| issue=1 \| pages=1–21}}</ref>		<ref name="K">{{Citation \| last1=Kassmann \| first1=Moritz \| title=A priori estimates for integro-differential operators with measurable kernels \| url=http://dx.doi.org/10.1007/s00526-008-0173-6 \| doi=10.1007/s00526-008-0173-6 \| year=2009 \| journal=Calculus of Variations and Partial Differential Equations \| issn=0944-2669 \| volume=34 \| issue=1 \| pages=1–21}}</ref>
			<ref name="lara2011regularity">{{Citation \| last1=Lara \| first1= Héctor Chang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of non local parabolic equations \| journal=Calculus of Variations and Partial Differential Equations \| year=2011 \| pages=1--34}}</ref>
	}}		}}