Calculus of Variations and Geometric Measure Theory
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R. Scala

Optimal estimates for the triple junction function and other surprising aspects of the area functional

created by scala on 15 Jun 2017

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Inserted: 15 jun 2017
Last Updated: 15 jun 2017

Year: 2017

Abstract:

We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function $u:B_1(O)\rightarrow\R^2$, a specific function which represents the first example of map whose graph area shows nonlocal effects. This is a map taking only three different values $\alpha,\beta,\gamma\in \R^2$ in three equal circular sectors of the unit radius ball $B_1(O)$. We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in \cite{BP} (and the corresponding upper bound for the relaxed area functional of the map $u$) is optimal. At the same time, we show by means of a counterexample that such construction is not optimal if we consider different domains than $B_1(O)$, which still contain the same discontinuity set of $u$ in $B_1(O)$. Such domains are obtained from $B_1(O)$ erasing part of interior of the sectors where $u$ is constant.


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