Regularity results for fully nonlinear integro-differential equations: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
== Interior regularity results == | |||
* 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"/>. | * 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"/>. | ||
Revision as of 17:35, 28 May 2011
Interior regularity results
- For general fully nonlinear integro-differential equations, interior $C^{1,\alpha}$ estimates can be proved in a variety of situations. The simplest assumption would be for a translation invariant uniformly elliptic equations with respect to the class of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 1 [1]. There are several other $C^{1,\alpha}$ estimates for variations of this situation (smooth coefficients, kernels close to the smoothness class, etc...) [2].
- A 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 $C^{s+\alpha}$ [3]. This is enough regularity for the solutions to be classical.
References
- ↑ 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
- ↑ 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
- ↑ Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X