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The relaxed area with respect to the strict convergence in BV

Simone Carano
Institution: 
SISSA
Schedule: 
Friday, October 22, 2021 - 14:00 to 15:30
Location: 
A-133
Location: 
Hybrid: in presence and online. Sign in to get the link to the webinar
Abstract: 

Many interesting problems in Calculus of Variations are connected to the relaxation of the area functional, which is a procedure that allows extending the notion of area of a graph to non-smooth functions. However, when the codimension is greater than 1, the problem of finding an integral representation of the relaxed area has a negative answer, even in the class of Sobolev functions. The reason is hidden in the non-local behaviour of the relaxed area functional with respect to the $L^1$ topology, as Acerbi and Dal Maso showed in the '90s.In our work, we propose a different approach, choosing the strict convergence in $BV$ as the topology of the relaxation. We will focus on the 2-dimensional and 2-codimensional case and in particular on functions valued in the unit circumference $\mathcal{S}^1$. We will give an explicit integral representation of the relaxed area on $W^{1,1}(\Omega;\mathcal{S}^1)$ and for some piecewise constant maps in $BV(\Omega;\mathcal{S}^1)$

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