Hölder estimates - Revision history

imported>Hector at 15:40, 8 July 2016

2016-07-08T15:40:55Z

← Older revision		Revision as of 10:40, 8 July 2016
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	<ref name="song2004">{{Citation \| last1=Song \| first1= Renming \| last2=Vondra\vcek \| first2= Zoran \| title=Harnack inequality for some classes of Markov processes \| url=http://dx.doi.org/10.1007/s00209-003-0594-z \| journal=Math. Z. \| issn=0025-5874 \| year=2004 \| volume=246 \| pages=177--202 \| doi=10.1007/s00209-003-0594-z}}</ref>		<ref name="song2004">{{Citation \| last1=Song \| first1= Renming \| last2=Vondra\vcek \| first2= Zoran \| title=Harnack inequality for some classes of Markov processes \| url=http://dx.doi.org/10.1007/s00209-003-0594-z \| journal=Math. Z. \| issn=0025-5874 \| year=2004 \| volume=246 \| pages=177--202 \| doi=10.1007/s00209-003-0594-z}}</ref>
	<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>		<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>
	<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor ~~Chang~~ \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>		<ref name="lara2012regularity">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</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>		<ref name="chang2014h">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Davila \| first2= Gonzalo \| title=Holder estimates for non-local parabolic equations with critical drift \| journal=arXiv preprint arXiv:1408.0676}}</ref>
	<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>		<ref name="lara2014regularity">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of non local parabolic equations \| journal=Calculus of Variations and Partial Differential Equations \| year=2014 \| volume=49 \| pages=139--172}}</ref>
	<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>		<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>
	<ref name="caffarelli2012holder">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=Holder regularity for generalized master equations with rough kernels}}</ref>		<ref name="caffarelli2012holder">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=Holder regularity for generalized master equations with rough kernels}}</ref>
	}}		}}

imported>Luis: /* Other variants */

2015-04-14T15:24:16Z

Other variants

← Older revision		Revision as of 10:24, 14 April 2015
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	* There are Holder estimates for equations in divergence form that are nonlocal in time <ref name="zacher2013" />, and also nonlocal in space-time together <ref name="caffarelli2012holder" />.		* There are Holder estimates for equations in divergence form that are nonlocal in time <ref name="zacher2013" />, and also nonlocal in space-time together <ref name="caffarelli2012holder" />.
	* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.		* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.
			* The interaction of an integral diffusion with first order terms brings some extra issues that are described in the page of [[drift-diffusion equations]].

	== References ==		== References ==

imported>Luis: /* Non variational equations */

2015-04-14T15:04:54Z

Non variational equations

← Older revision		Revision as of 10:04, 14 April 2015
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	* Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part <ref name="lara2012regularity" />. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.<ref name="lara2014regularity" />. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels <ref name="chang2014h" />. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).		* Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part <ref name="lara2012regularity" />. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.<ref name="lara2014regularity" />. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels <ref name="chang2014h" />. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
	* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.		* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
	* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].		* A generalization of all previous results (except for variable order) was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].
	* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not Holder.		* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not Holder.

imported>Luis: /* Non variational equations */

2015-04-10T21:59:01Z

Non variational equations

← Older revision		Revision as of 16:59, 10 April 2015
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	* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.		* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
	* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].		* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].
	* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not ~~H\"older~~.		* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not Holder.

	== Other variants ==		== Other variants ==

imported>Luis at 21:58, 10 April 2015

2015-04-10T21:58:38Z

← Older revision		Revision as of 16:58, 10 April 2015
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	== Other variants ==		== Other variants ==

	* There are Holder estimates for equations in divergence form that are ~~non local~~ in time <ref name="zacher2013" />		* There are Holder estimates for equations in divergence form that are nonlocal in time <ref name="zacher2013" />, and also nonlocal in space-time together <ref name="caffarelli2012holder" />.
	* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.		* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.

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	<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>		<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>
	<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>		<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>
			<ref name="caffarelli2012holder">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=Holder regularity for generalized master equations with rough kernels}}</ref>
	}}		}}

imported>Luis: /* Non variational equations */

2015-04-10T21:52:58Z

Non variational equations

← Older revision		Revision as of 16:52, 10 April 2015
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	* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.		* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
	* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].		* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].
	* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity which is not H\"older.		* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not H\"older.

	== Other variants ==		== Other variants ==

imported>Luis: /* Non variational equations */

2015-04-10T21:52:17Z

Non variational equations

← Older revision		Revision as of 16:52, 10 April 2015
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	* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.		* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
	* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].		* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels\|here]].
	* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like.		* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $\|y\|$ which is not necessarily power-like. For some kernels of the form $K(x,y) \approx \|y\|^{-n}$, they obtain an estimate of a modulus of continuity which is not H\"older.

	== Other variants ==		== Other variants ==

imported>Luis: /* Non variational equations */

2015-04-08T22:03:53Z

Non variational equations

← Older revision		Revision as of 17:03, 8 April 2015
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	* A result by Bass and Kassmann also uses probabilistic techniques.<ref		* A result by Bass and Kassmann also uses probabilistic techniques.<ref
	name="BK"/> It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.		name="BK"/> It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
	* The first purely analytic proof was obtained by ~~Luis~~ Silvestre <ref		* The first purely analytic proof was obtained by Silvestre <ref
	name="S"/>. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.		name="S"/>. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
	* The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre <ref name="CS"/>. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).		* The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre <ref name="CS"/>. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).

imported>Luis: /* References */

2015-04-08T22:02:45Z

References

← Older revision		Revision as of 17:02, 8 April 2015
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	<ref name="zacher2013">{{Citation \| last1=Zacher \| first1= Rico \| title=A De Giorgi--Nash type theorem for time fractional diffusion equations \| url=http://dx.doi.org/10.1007/s00208-012-0834-9 \| journal=Math. Ann. \| issn=0025-5831 \| year=2013 \| volume=356 \| pages=99--146 \| doi=10.1007/s00208-012-0834-9}}</ref>		<ref name="zacher2013">{{Citation \| last1=Zacher \| first1= Rico \| title=A De Giorgi--Nash type theorem for time fractional diffusion equations \| url=http://dx.doi.org/10.1007/s00208-012-0834-9 \| journal=Math. Ann. \| issn=0025-5831 \| year=2013 \| volume=356 \| pages=99--146 \| doi=10.1007/s00208-012-0834-9}}</ref>
	<ref name="barles2011">{{Citation \| last1=Barles \| first1= Guy \| last2=Chasseigne \| first2= Emmanuel \| last3=Imbert \| first3= Cyril \| title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations \| url=http://dx.doi.org/10.4171/JEMS/242 \| journal=J. Eur. Math. Soc. (JEMS) \| issn=1435-9855 \| year=2011 \| volume=13 \| pages=1--26 \| doi=10.4171/JEMS/242}}</ref>		<ref name="barles2011">{{Citation \| last1=Barles \| first1= Guy \| last2=Chasseigne \| first2= Emmanuel \| last3=Imbert \| first3= Cyril \| title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations \| url=http://dx.doi.org/10.4171/JEMS/242 \| journal=J. Eur. Math. Soc. (JEMS) \| issn=1435-9855 \| year=2011 \| volume=13 \| pages=1--26 \| doi=10.4171/JEMS/242}}</ref>
	<ref name="rang2013h">{{Citation \| last1=Rang \| first1= Marcus \| last2=Kassmann \| first2= Moritz \| last3=Schwab \| first3= Russell W \| title=~~H$\backslash$" older~~ Regularity For Integro-Differential Equations With Nonlinear Directional Dependence \| journal=arXiv preprint arXiv:1306.0082}}</ref>		<ref name="rang2013h">{{Citation \| last1=Rang \| first1= Marcus \| last2=Kassmann \| first2= Moritz \| last3=Schwab \| first3= Russell W \| title=Holder Regularity For Integro-Differential Equations With Nonlinear Directional Dependence \| journal=arXiv preprint arXiv:1306.0082}}</ref>
	<ref name="bogdan2005harnack">{{Citation \| last1=Bogdan \| first1= Krzysztof \| last2=Sztonyk \| first2= Pawe\l \| title=Harnack’s inequality for stable Lévy processes \| journal=Potential Analysis \| year=2005 \| volume=22 \| pages=133--150}}</ref>		<ref name="bogdan2005harnack">{{Citation \| last1=Bogdan \| first1= Krzysztof \| last2=Sztonyk \| first2= Pawe\l \| title=Harnack’s inequality for stable Lévy processes \| journal=Potential Analysis \| year=2005 \| volume=22 \| pages=133--150}}</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>
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	<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>		<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>
	<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor Chang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>		<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor Chang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>
	<ref name="chang2014h">{{Citation \| last1=Chang-Lara \| first1= Hector \| last2=Davila \| first2= Gonzalo \| title=~~H$\backslash$" older~~ estimates for non-local parabolic equations with critical drift \| journal=arXiv preprint arXiv:1408.0676}}</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>
	<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>		<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>
	<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>		<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>
	}}		}}

imported>Luis: /* Other variants */

2015-04-08T22:01:22Z

Other variants

← Older revision		Revision as of 17:01, 8 April 2015
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	* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />		* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
	* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a ~~very general~~ class of integral equations <ref name="barles2011" />.		* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.


	== References ==		== References ==

← Older revision		Revision as of 10:40, 8 July 2016
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	<ref name="song2004">{{Citation \| last1=Song \| first1= Renming \| last2=Vondra\vcek \| first2= Zoran \| title=Harnack inequality for some classes of Markov processes \| url=http://dx.doi.org/10.1007/s00209-003-0594-z \| journal=Math. Z. \| issn=0025-5874 \| year=2004 \| volume=246 \| pages=177--202 \| doi=10.1007/s00209-003-0594-z}}</ref>		<ref name="song2004">{{Citation \| last1=Song \| first1= Renming \| last2=Vondra\vcek \| first2= Zoran \| title=Harnack inequality for some classes of Markov processes \| url=http://dx.doi.org/10.1007/s00209-003-0594-z \| journal=Math. Z. \| issn=0025-5874 \| year=2004 \| volume=246 \| pages=177--202 \| doi=10.1007/s00209-003-0594-z}}</ref>
	<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>		<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>
	<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor ~~Chang~~ \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>		<ref name="lara2012regularity">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</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>		<ref name="chang2014h">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Davila \| first2= Gonzalo \| title=Holder estimates for non-local parabolic equations with critical drift \| journal=arXiv preprint arXiv:1408.0676}}</ref>
	<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>		<ref name="lara2014regularity">{{Citation \| last1=Chang-Lara \| first1= Héctor \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of non local parabolic equations \| journal=Calculus of Variations and Partial Differential Equations \| year=2014 \| volume=49 \| pages=139--172}}</ref>
	<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>		<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>
	<ref name="caffarelli2012holder">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=Holder regularity for generalized master equations with rough kernels}}</ref>		<ref name="caffarelli2012holder">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=Holder regularity for generalized master equations with rough kernels}}</ref>
	}}		}}

← Older revision		Revision as of 17:02, 8 April 2015
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	<ref name="zacher2013">{{Citation \| last1=Zacher \| first1= Rico \| title=A De Giorgi--Nash type theorem for time fractional diffusion equations \| url=http://dx.doi.org/10.1007/s00208-012-0834-9 \| journal=Math. Ann. \| issn=0025-5831 \| year=2013 \| volume=356 \| pages=99--146 \| doi=10.1007/s00208-012-0834-9}}</ref>		<ref name="zacher2013">{{Citation \| last1=Zacher \| first1= Rico \| title=A De Giorgi--Nash type theorem for time fractional diffusion equations \| url=http://dx.doi.org/10.1007/s00208-012-0834-9 \| journal=Math. Ann. \| issn=0025-5831 \| year=2013 \| volume=356 \| pages=99--146 \| doi=10.1007/s00208-012-0834-9}}</ref>
	<ref name="barles2011">{{Citation \| last1=Barles \| first1= Guy \| last2=Chasseigne \| first2= Emmanuel \| last3=Imbert \| first3= Cyril \| title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations \| url=http://dx.doi.org/10.4171/JEMS/242 \| journal=J. Eur. Math. Soc. (JEMS) \| issn=1435-9855 \| year=2011 \| volume=13 \| pages=1--26 \| doi=10.4171/JEMS/242}}</ref>		<ref name="barles2011">{{Citation \| last1=Barles \| first1= Guy \| last2=Chasseigne \| first2= Emmanuel \| last3=Imbert \| first3= Cyril \| title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations \| url=http://dx.doi.org/10.4171/JEMS/242 \| journal=J. Eur. Math. Soc. (JEMS) \| issn=1435-9855 \| year=2011 \| volume=13 \| pages=1--26 \| doi=10.4171/JEMS/242}}</ref>
	<ref name="rang2013h">{{Citation \| last1=Rang \| first1= Marcus \| last2=Kassmann \| first2= Moritz \| last3=Schwab \| first3= Russell W \| title=~~H$\backslash$" older~~ Regularity For Integro-Differential Equations With Nonlinear Directional Dependence \| journal=arXiv preprint arXiv:1306.0082}}</ref>		<ref name="rang2013h">{{Citation \| last1=Rang \| first1= Marcus \| last2=Kassmann \| first2= Moritz \| last3=Schwab \| first3= Russell W \| title=Holder Regularity For Integro-Differential Equations With Nonlinear Directional Dependence \| journal=arXiv preprint arXiv:1306.0082}}</ref>
	<ref name="bogdan2005harnack">{{Citation \| last1=Bogdan \| first1= Krzysztof \| last2=Sztonyk \| first2= Pawe\l \| title=Harnack’s inequality for stable Lévy processes \| journal=Potential Analysis \| year=2005 \| volume=22 \| pages=133--150}}</ref>		<ref name="bogdan2005harnack">{{Citation \| last1=Bogdan \| first1= Krzysztof \| last2=Sztonyk \| first2= Pawe\l \| title=Harnack’s inequality for stable Lévy processes \| journal=Potential Analysis \| year=2005 \| volume=22 \| pages=133--150}}</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>
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	<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>		<ref name="silvestre2011differentiability">{{Citation \| last1=Silvestre \| first1= Luis \| title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion \| journal=Advances in mathematics \| year=2011 \| volume=226 \| pages=2020--2039}}</ref>
	<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor Chang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>		<ref name="lara2012regularity">{{Citation \| last1=Lara \| first1= Héctor Chang \| last2=Dávila \| first2= Gonzalo \| title=Regularity for solutions of nonlocal, nonsymmetric equations \| year=2012 \| volume=29 \| pages=833--859}}</ref>
	<ref name="chang2014h">{{Citation \| last1=Chang-Lara \| first1= Hector \| last2=Davila \| first2= Gonzalo \| title=~~H$\backslash$" older~~ estimates for non-local parabolic equations with critical drift \| journal=arXiv preprint arXiv:1408.0676}}</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>
	<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>		<ref name="lara2014regularity">{{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=2014 \| volume=49 \| pages=139--172}}</ref>
	<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>		<ref name="bjorland2012">{{Citation \| last1=Bjorland \| first1= C. \| last2=Caffarelli \| first2= L. \| last3=Figalli \| first3= A. \| title=Non-local gradient dependent operators \| url=http://dx.doi.org/10.1016/j.aim.2012.03.032 \| journal=Adv. Math. \| issn=0001-8708 \| year=2012 \| volume=230 \| pages=1859--1894 \| doi=10.1016/j.aim.2012.03.032}}</ref>
	}}		}}