16:30
So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the ’80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding problem, open since the ‘70s, of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and provide a complete regularity theory for a new double phase model. From joint work with Giuseppe Mingione (University of Parma).