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| == Interior regularity results ==
| | #REDIRECT [[Linear integro-differential operator]] |
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| * 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"/>.
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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 solutions|classical]].
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| == References ==
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| {{reflist|refs=
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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>
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| <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>
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| <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>
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| [[Category:Fully nonlinear equations]]
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Latest revision as of 11:40, 29 May 2011