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
modified on 17 Jan 2018


in revision

Inserted: 15 jun 2017
Last Updated: 17 jan 2018

Year: 2017


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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