C^1 regularity for degenerate elliptic equations in the plane

par Thibault Lacombe (Université de Toulouse)

mardi 20 mai 2025, 14:00 → 16:00 Europe/Paris

In this presentation, we focus on the regularity of solutions to the PDE div(G(Du)) = 0 in two dimensions. We show that Lipschitz solutions to such equations are of class C^1 under a mild ellipticity assumption on the field G. This result generalizes those obtained by DeSilva and Savin (Duke, 2010) in the variational case G = DF, with F strictly convex. In the general case, we demonstrate that the definition of the degenerate set associated with G must be reinterpreted, and we highlight the effect—absent in the non-degenerate case—of the antisymmetric part of DG. This is a joint work with X.Lamy (Toulouse).