Some basic facts on the system \Delta u - W_u (u) = 0

TL;DR

This paper reformulates a nonlinear PDE system involving the Laplacian and a potential term as a divergence-free stress-energy tensor, highlighting differences between phase-transition and Ginzburg-Landau models.

Contribution

It introduces a new formulation of the system using a stress-energy tensor and analyzes key differences between phase-transition and Ginzburg-Landau paradigms.

Findings

Reformulation as div T = 0 for the system

Identification of differences between phase-transition and Ginzburg-Landau systems

Derivation of a priori properties of solutions

Abstract

We rewrite the system \Delta u - W_u (u) = 0, for u: R^n to R^n, in the form 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.