SCHEDULE
Introduction
…
lecturers
Elliptic PDEs: recent extensions on Hilbert’s 19th problem
Xavier Cabré
ICREA – UAB – CRM
Abstract
Serendipity strikes again
Joaquim Ortega
UB – CRM
Xavier Tolsa
ICREA – UAB – CRM
SCHEDULE
15:00 – 15:10 | Welcome |
15:10 – 15:50 | TBA Xavier Tolsa ICREA - UAB - CRM |
16:00 – 16:30 | Coffee Break |
16:30 – 17:10 | Serendipity strikes again Joaquim Ortega UB - CRM |
17:20 – 18:00 | Elliptic PDEs: recent extensions on Hilbert’s 19th problem Xavier Cabré ICREA - UPC - CRM |
LIST OF PARTICIPANTS
| Name | Institution |
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Elliptic PDEs: recent extensions on Hilbert’s 19th problem
Abstract:
In 1900 Hilbert’s 19th problem asked whether minimizers of elliptic functionals are always analytic. A celebrated result of De Giorgi and Nash in the late 1950s solved the problem for uniformly convex (equivalently, uniformly elliptic) functionals. The important case of the minimal surfaces equation required substantial additional work, culminated by Simons 1968 paper.
We will explain such developments and then turn into a closely related problem: the regularity of stable solutions, a larger class than absolute minimizers. In this case, positive answers have only been established in the current 2020 decade, though some cases remain still open. For stable minimal surfaces, the results are proved in several articles by Chodosh, Li, Minter, Stryker, and Mazet. For reaction-diffusion equations, the optimal result has been established in a paper by Cabré, Figalli, Ros-Oton, and Serra. We will finally mention some open problems on the regularity of stable solutions -the main ones being the regularity of stable minimal surfaces in R^7 and of stable solutions to reaction equations for the fractional Laplacian.
Xavier Cabré (ICREA & UPC & CRM)