Luis Caffarelli
Core Contributions
Caffarelli is a towering figure in the field of partial differential equations and free boundary problems, and the founding father of regularity theory for the fractional Laplacian. He systematically extended the classical second-order elliptic regularity theory to nonlocal equations, establishing Schauder estimates, Harnack inequalities, and the extension formula for fractional equations.
Selected Works
| Year | Work / Article | Problem Addressed |
|---|---|---|
| 2007 | "An extension problem related to the fractional Laplacian" (with Silvestre) | Established the extension formula for the fractional Laplacian, converting nonlocal problems into local degenerate elliptic equations and providing a core tool for subsequent research |
| 2009 | "Regularity theory for fully nonlinear integro-differential equations" (with Silvestre) | Established the regularity theory for fully nonlinear nonlocal integro-differential equations, including ABP estimates, Krylov-Safonov estimates, and $C^{2s+\alpha}$ interior estimates |
| 2011 | "Regularity theory for parabolic nonlinear integral operators" (with Chan and Vasseur) | Extended the nonlocal regularity theory to the parabolic setting, establishing the De Giorgi-Nash-Moser theory for nonlocal heat equations |