Upper bounds for the relaxed area of S¹-valued Sobolev maps and its countably subadditive interior envelope

Given a connected bounded open Lipschitz set Ω ⊂ R² , we show that the relaxed Cartesian area functional Ā(u, Ω) of a map u ∈ W1,1(Ω;S¹) is finite, and we provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture adapted to W1,1(Ω; S¹) , on the largest countably subadditive set function {Ā}(u, ·) smaller than or equal to Ā(u, ·) .