Counterexamples to Modica's gradient estimate for systems arising in multi-phase transitions

TL;DR

This paper constructs numerous one-dimensional periodic solutions for certain elliptic systems that violate Modica's gradient estimate, addressing open problems and extending previous counterexamples in the study of phase transition models.

Contribution

It provides new explicit counterexamples to Modica's gradient estimate for elliptic systems, advancing understanding of phase transition models and related mathematical conjectures.

Findings

Constructed many periodic solutions violating Modica's estimate

Extended previous counterexamples in elliptic systems

Addresses open problems in phase transition theory

Abstract

We construct a plethora of one-dimensional periodic solutions for a class of semilinear elliptic systems of phase transition type which violate Modica's gradient estimate. This complements a recent technical counterexample in arXiv:1401.4847 and is partly motivated by a recent open problem in arXiv:1506.02731.