A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips

Abstract

Given a bounded Lipschitz domain ω ⊂ R d−1 and a lower semicontinuous function W : R N → R+ ∪ {+∞} that vanishes on a finite set and that is bounded from below by a positive constant at infinity, we show that every map u : R × ω → R N with R×ω |∇u| 2 + W (u) dx1 dx ′ < +∞ has a limit u ± ∈ {W = 0} as x1 → ±∞. The convergence holds in L 2 (ω) and almost everywhere in ω. We also prove a similar result for more general potentials W in the case where the considered maps u are divergence-free in Ω with ω being the (d − 1)-torus and N = d.