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(\Omega,S^2)\mapsto\mathbb{R}_+$ defined by

$\displaystyle E_w(u)=\mathop{\text{Inf}}\bigg\{\liminf_{n\to\infty}\int_\Omega\... ...^1(\overline\Omega,S^2),\;u_n\rightharpoonup u\;\,\text{weakly in $H^1$}\bigg\}$

where $ \Omega$ is a smooth bounded domain in $ \mathbb{R}^3$ and $ w:\Omega\to\mathbb{R}_+$ is a measurable function taking values between two positive constants a.e. in $ \Omega$. The explicit formula involves a distance function $ d_w$ on $ \overline\Omega$ 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.