An Epiperimetric Approach to Isolated Singularities

Max Engelstein (University of Minnesota)

18-Jun-2020, 14:00-14:50 (4 years ago)

Abstract: The presence of singular points (i.e. points around which the object in question does not look flat at any scale) is inevitable in most minimization problems. One fundamental question is whether minimizers have a unique tangent object at singular points i.e., is the minimizer increasingly well approximated by some other minimizing object as we “zoom in” at a singular point. This question has been investigated with varying degrees of success in the settings of minimal surfaces, harmonic maps and obstacle problems amongst others.

In this talk, we will present an uniqueness of blowups result for minimizers of the Alt-Caffarelli functional. In particular, we prove that the tangent object is unique at isolated singular points in the free boundary. Our main tool is a new approach to proving (log-)epiperimetric inequalities at isolated singularities. This epiperimetric inequality differs from previous ones in that it holds without any additional assumptions on the symmetries of the tangent object.

If we have time, we will also discuss how this method allows us to recover some uniqueness of blow-ups results in the minimal surfaces setting, particularly those of Allard-Almgren (’81) and Leon Simon (’83). This is joint work with Luca Spolaor (UCSD) and Bozhidar Velichkov (U. Napoli).

analysis of PDEs

Audience: researchers in the topic

Comments: https://nguyenquochung1241.wixsite.com/qhung/post/pde-seminar-via-zoom


PDE seminar via Zoom

Organizer: Quoc-Hung Nguyen*
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