IUMJ

Title: Relaxed energies for $H^{1/2}$-maps with values into the circle and measureable weights

Authors: Vincent Millot and Adriano Pisante

Issue: Volume 58 (2009), Issue 1, 49-136

Abstract:

We consider, for maps $f\in\dot{H}^{1/2}(\mathbb{R}^2;\mathbb{S}^1)$, an energy $\mathcal{E}(f)$ related to a seminorm equivalent to the standard one. This seminorm is associated to a measurable matrix field in the half space. Under structure assumptions on it, we show that the infimum of $\mathcal{E}$ over a class of maps with two prescribed singularities induces a natural geodesic distance on the plane. In case of a continuous matrix field, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a geodesic curve on the plane. We describe this concentration in terms of bubbling-off of circles along this curve. Then we explicitly compute the relaxation with respect to the weak $\dot{H}^{1/2}$-convergence of the functional $f\mapsto\mathcal{E}(f)$ if $f$ is smooth and $+\infty$ otherwise. The formula involves the length of a minimal connection between the singularities of $f$ computed in terms of the distance previously obtained.