Event description
In this talk, I will address a question raised by D. De Silva and O. Savin concerning the regularity of planar solutions to div(Df(Du)) = 0 under the sole assumption that f is C^1 is strictly convex and u is Lipschitz continuous. After a brief overview of the literature on such highly degenerate equations, starting from the foundational work of De Silva and Savin, I will focus on new regularity results obtained in collaboration with X. Lamy. These show, essentially, that u is C^1 regular up to an isolated set of points provided f is fully degenerate only along C^1 curves, an extension of the previously known results that required f to be fully degenerate only at isolated points. I will explain some of the main ideas of the work, in particular the connection of this problem with Hamilton-Jacobi equations, and, if time allows, some details of the arguments.
This conference is funded by the European Research Council (ERC) under the Horizon Europe research and innovation programme (grant agreement No. 101220121 - project: NEW - Nonuniform Ellipticity Widespread).