##### Department of Mathematics,

University of California San Diego

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### Special Geometric Analysis Seminar

## Luca Spolaor

#### Massachusetts Institute of Technology

## (Log)-Epiperimetric Inequality and the Regularity of Free-Boundaries

##### Abstract:

In this talk I will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. Using as a model case the so-called Obstacle problem, I will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (Log) -epiperimetric inequality. This technique is very general, and much like Caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. Moreover it is able to deal with logarithmic regularity, which in the case of the Obstacle problem is optimal due to an example of Figalli-Serra. If time permits I will explain how such an inequality is linked to the behavior of a gradient flow at infinity.

Host: Bennet Chow

### November 29, 2018

### 1:00 PM

### AP&M 7218

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