Bellman equation and Regularity results for fully nonlinear integro-differential equations: Difference between pages

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The Bellman equation is the equality
== Interior regularity results ==
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.


The equation appears naturally in problems of stochastic control with [[Levy processes]].
* 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"/>.


The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
* 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]].


Note that any '''convex''' fully nonlinear elliptic PDE of second order $F(D^2u, Du, u, x)$ can be written as a Bellman equation by taking the supremum of all supporting planes of $F$. It is not known whether that such representation holds for integro-differential equations.
== 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>
}}


[[Category:Fully nonlinear equations]]
[[Category:Fully nonlinear equations]]

Revision as of 01:24, 2 June 2011

Interior regularity results

References

  1. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  2. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity results for nonlocal equations by approximation", Archive for Rational Mechanics and Analysis (Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527 
  3. Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X