| Contents |
| We study the action functional
\[
\int_{\mathbb{R}\times \omega} 1 - \sqrt{1 - |\nabla u|^{2}} + W(u) \ d x.
\]
We assume that $\omega\subset\mathbb{R}^{N-1}$ is a bounded domain and that $W\colon\mathbb{R}\to[0,+\infty[$ is a double-well potential, i.e., $W$ is of class $C^1$ and satisfies $W(-1) = W(1) = 0$ and $W(u)>0$ if $u\ne \pm 1$.
Using variational arguments and a rearrangement technique, we prove the existence, one-dimensionality and uniqueness (up to translations) of a smooth minimizing phase transition between the stable states $-1$ and $1$.
We also discuss the existence of minimizing heteroclinic connections for the non autonomous model
\[
\int_{\mathbb{R}} 1 - \sqrt{1 - |u'|^{2}} + a(t)W(u) \ d t,
\]
where $a\in C^1(\mathbb{R})$ is a bounded positive function that can have a constant asymptotic behaviour at infinity or can be periodic.
The Euler-Lagrange equations associated with these functionals find significant applications in Riemannian geometry and in the theory of relativity from where timelike heteroclinic solutions take their name. |
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