Edgard Pimentel - Regularity Theory for Nonlinear PDEs

Explore recent developments in regularity theory for nonlinear PDEs in this 49-minute lecture by Edgard Pimentel, presented as part of the Hausdorff Trimester Program on Evolution of Interfaces. Delve into a two-layered approach that connects complex problems to those with established theories, utilizing compactness and stability concepts for approximation, followed by scaling arguments for localized analysis. Examine toy models including fully nonlinear PDEs, the Isaacs equation, double-divergence problems, and degenerate/singular equations. Progress through topics such as existence of small correctors, oscillation control, non-variational settings, Hölder regularity of gradients, and approximation lemmas. Conclude with an exploration of free boundary problems, gaining insights into sharp regularity results and analytical developments in this field.

Intro
General overview
Motivation applications
Motivation analytical developments
A few questions
A warm up result
A warm up proof
Strategy of the proof
Existence of small correctors
Oscillation control along the critical set
Non-variational setting
A detour on the case of constant exponents
Hölder regularity of the gradient
Preliminary levels of compactness
Approximation Lemma
Prool of the theorem - Induction argument