Relaxed Energies for $ H^{1/2}$-Maps with Values into the Circle and Measureable Weights


Vincent Millot
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
vmillot@andrew.cmu.edu

and

Adriano Pisante
Department of Mathematics
Univeristy of Roma 'La Sapienza'
P.le Allo Moro 5
00185 Roma, Italy
pisante@mat.uniroma1.it

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. We show that the infimum of $ \mathcal{E}$ over a class of maps having finitely many prescribed singularities, is equal to the length of a minimal connection between the singularities with respect to 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 1-rectifiable current on the plane. We describe this concentration in terms of bubbling-off of circles. 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.

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  • 06-CNA-009.pdf