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{\bf Vincent Millot}\\
Carnegie Mellon University\\
{\tt vmillot@andrew.cmu.edu}
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{\it Relaxed Energies for $S^2$--Valued Maps, Nonhomogeneous Distances and Minimal Connections}

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\noindent {\bf Abstract}:  In this work, we compute the functional $E_w:H^1(\Omega,S^2)\mapsto\mathbb{R}_+$ defined by 

$$E_w(u)=\mathop{\text{Inf}}\bigg\{\liminf_{n\to\infty}\int_\Omega|\nabla u|^2w(x)dx\,;\;u_n\in C^1(\overline\Omega,S^2),\;u_n\rightharpoonup u\;\,\text{weakly in $H^1$}\bigg\}$$

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.


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