Regularity results for nonlocal equations - Revision history

imported>Luis at 01:45, 25 April 2012

2012-04-25T01:45:41Z

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	* A [[nonlocal Evans-Krylov theorem\|nonlocal version of Evans-Krylov theorem]] says that for the [[Bellman equation]], for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough regularity for the solutions to be classical.		* A [[nonlocal Evans-Krylov theorem\|nonlocal version of Evans-Krylov theorem]] says that for the [[Bellman equation]], for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough regularity for the solutions to be classical.

			== 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="CS2">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Silvestre \| first2=Luis \| title=Regularity results for nonlocal equations by approximation \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| year=2009 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=1–30}}</ref>
			<ref name="CS3">{{Citation \| last1=Caffarelli \| first1=Luis \| last2=Silvestre \| first2=Luis \| title=The Evans-Krylov theorem for non local fully non linear equations \| year=to appear \| journal=[[Annals of Mathematics]] \| issn=0003-486X}}</ref>
			}}

imported>Luis: Created page with "{{stub}} == Regularity results for linear equations with constant coefficients == == Regularity results for linear equations with smooth coefficients == == Regularity results ..."

2012-04-25T01:45:15Z

Created page with "{{stub}} == Regularity results for linear equations with constant coefficients == == Regularity results for linear equations with smooth coefficients == == Regularity results ..."

New page

{{stub}}

== Regularity results for linear equations with constant coefficients ==

== Regularity results for linear equations with smooth coefficients ==

== Regularity results for linear equations with rough coefficients ==

== Regularity results for nonlinear equations ==

* For general [[fully nonlinear integro-differential equations]], interior [[differentiability estimates|$C^{1,\alpha}$ estimates]] can be proved in a variety of situations. The simplest assumption would be for a translation invariant [[fully nonlinear integro-differential equations|uniformly elliptic]] equations with respect to the [[Linear integro-differential operator|class of kernels]] that are uniformly elliptic of order $s$ and in the smoothness class of order 1 <ref name="CS"/>. There are several other [[differentiability estimates|$C^{1,\alpha}$ estimates]] for variations of this situation (smooth coefficients, kernels close to the smoothness class, etc...) <ref name="CS2"/>.

* A [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]] says that for the [[Bellman equation]], for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ <ref name="CS3"/>. This is enough regularity for the solutions to be classical.