Seminari del Dipartimento di Matematica

Seminario del 2019

Daniela De Silva

Viscosity solutions approach to variational problems

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

In this talk we discuss some extensions of the classical Krylov-Safonov Harnack inequality. After reviewing the standard regularity theory, we will introduce a weaker notion of viscosity solutions. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. Roughly, our viscosity solutions satisfy comparison in a neighborhood of a touching point whose size depends on the properties of the test functions. As an application, we recover the C^{1,\alpha} estimates of Almgren and Tamanini for quasi-minimizers of the perimeter functional. We also establish the regularity of the free boundary for almost minimizers of one-phase type problems.