Harmonic Analysis Techniques for -Almost- Minimizers

Explore harmonic analysis techniques for (almost-)minimizers in this 45-minute lecture from the Hausdorff Center for Mathematics. Delve into the intersection of free boundary theory, calculus of variations, and geometric measure theory as they apply to the Alt-Caffarelli functionals. Examine the relationship between harmonic measure behavior and domain geometry, and discover how these tools inform the study of one and two-phase problems. Learn about green functions, free boundary problems, and their applications to harmonic measure. Investigate the concept of almost-minimizers, their compactness properties, and the regularity of free boundaries. Gain insights into proof techniques for connectedness and Ahlfors regularity. Explore branch points in two-phase problems and consider open questions in the field. Based on collaborative work with Guy David, Mariana Smit Vega Garcia, and Tatiana Toro, this talk offers a comprehensive look at cutting-edge research in harmonic analysis and geometric measure theory.

Intro
TALK PREVIEW
GREEN FUNCTION AND FREE BOUNDARY PROBLEM
APPLICATIONS TO HARMONIC MEASURE
WHY ALMOST-MINIMIZERS
ALMOST-MINIMIZERS AND COMPACTNESS
REGULARITY OF THE FREE BOUNDARY
PROOF IDEAS PART II: CONNECTEDNESS
PROOF IDEAS PART III: AHLFORS REGULARITY
TWO-PHASE PROBLEM: BRANCH POINTS
BUT DO BRANCH POINTS ACTUALLY HAPPEN?
BIG OPEN PROBLEM