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An introduction to surfaces and varifolds with free boundaries and their structure

Luigi De Masi
Friday, June 12, 2020 - 14:00
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Surfaces with free boundary naturally arise as solutions of variational problems where one has to find, in a fixed domain $\mathcal{M} \subset \mathbb{R}^n$, a $k$-surface $\Sigma$ which is stationary for some energy, and whose boundary $\partial \Sigma$ is free to move on the boundary $\partial \mathcal{M}$ of the ambient domain.

In this talk I will present the classical notion of surface with free boundary and some basic properties. After this introduction, I will introduce the concept of varifold as weak notion of surface, and the idea of varifolds with free boundaries as weak solutions to free boundary problems.

Next I will discuss some questions on which I am working and some results on the structure of the free boundary: we will show that there is a bound on the "length" of the free boundary for general $k$-varifolds and, under suitable assumptions, that the free boundary is $(k-1)$-rectifiable.

The prerequisites in order to follow the presentation are basic notions of functional analysis, measure theory and differential geometry. 

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