Wolf Prize Laureate in Mathematics 2012
Affiliation at the time of the award:
University of Texas, USA
“for outstanding, visionary, original and fundamental work on partial differential equations, in particular on regularity for elliptic and parabolic equations, free boundary problems and fluid mechanics”.
Luis Caffarelli has repeatedly made very deep breakthroughs. His early work on free boundary problems was the first place where his extraordinary talent and intuition began to show. Free boundary problems are about finding both the solutions to an equation and the region where the equation holds. In a series of pioneering papers, Caffarelli put forward a novel methodology which eventually leads, after several truly amazing technical estimates that step by step to improve the regularity of the solutions and the boundary, to full regularity under very mild assumptions. Although the theory is complicated the arguments are elementary and full of beautiful geometric intuition and mastery of analytic technique.
A second fundamental contribution by Cafarelli is the study of fully nonlinear elliptic partial differential equations (including the famous Monge-Ampere equation), which he revolutionized. The upshot is that, although the equations are nonlinear, they behave for purposes of regularity as if they were linear. (Work of Nirenberg, Spruck, Evans, Krylov and others also played a significant role here.)
Another fundamental contribution by Cafarelli is his joint work with Kohn and Nirenberg on partial regularity of solutions of the incompressible Navier-Stokes equation in 3 space dimensions. Although the full regularity of solutions is still unknown and likely very hard, Caffarelli-Kohn-Nirenberg showed that the singular set must have parabolic Hausdorff dimension strictly less than one. In particular, singular fibers cannot occur (V.Scheffer also deserves partial credit).
Prof. Caffarelli has also produced deep work on homogenization and on equations with nonlocal dissipation. The list could be continued. Caffarelli is the world’s leading expert on regularity of solutions of partial differential equations.