Some basic facts on the system Δ𝑢-𝑊ᵤ(𝑢)=0

Alikakos, Nicholas

doi:10.1090/S0002-9939-2010-10453-7

Some basic facts on the system $\Delta u - W_u (u) = 0$
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Proc. Amer. Math. Soc. 139 (2011), 153-162 Request permission

Abstract:

We rewrite the system $\Delta u - W_u (u) = 0$, for $u: \mathbb R^n \to \mathbb R^n$, in the form $\operatorname {div}T = 0$, where $T$ is an appropriate stress-energy tensor, and derive certain a priori consequences on the solutions. In particular, we point out some differences between two paradigms: the phase-transition system, with target a finite set of points, and the Ginzburg–Landau system, with target a connected manifold.

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