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

We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function u : B1(O) → R, 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 α, β, γ ∈ R in three equal circular sectors of the unit radius ball B1(O). We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in [5] (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 B1(O), which still contain the same discontinuity set of u in B1(O). Such domains are obtained from B1(O) erasing part of interior of the sectors where u is constant.