Vincent Millot Carnegie Mellon University

                    vmillot@andrew.cmu.edu

Relaxed Energies for S^{2}–Valued Maps, Nonhomogeneous Distances and Minimal Connections

Abstract: In this work, we compute the functional E_{w} : H^{1}(Ω, S^{2}) ↦ ℝ_{+} defined by

E_{w}(u) = Inf \{ lim inf_{n→∞} ∫_{Ω} |∇u|^{2}w(x)dx; u_{n} ∈ C^{1}(\overline{Ω}, S^{2}), u_{n} ⇀ u weakly in H^{1}\}

where Ω is a smooth bounded domain in ℝ^{3} and w : Ω → ℝ_{+} is a measurable function taking values between two positive constants a.e. in Ω. The explicit formula involves a distance function d_{w} on \overline{Ω} associated in a canonical way to w, and the notion of length of a minimal connection connecting the topological singularities of u. Stability results with respect to perturbations of the function w are also presented.