A class of nonlinear elasticity problems with no local but many global minimizers

TL;DR

This paper introduces a class of nonlinear elasticity models with incompatible energy wells, revealing that while many global minimizers exist, strong local minimizers are absent, and provides a new proof technique based on a nonlinear Clapeyron theorem.

Contribution

It establishes a novel class of elasticity problems with many global but no local minimizers and develops a new proof approach using a nonlinear Clapeyron theorem.

Findings

Existence of many global minimizers in the models.

Absence of strong local minimizers in the models.

A new proof method for affine boundary conditions.

Abstract

We present a class of models of elastic phase transitions with incompatible energy wells in any space dimension, where an abundance of Lipschitz global minimizers in a hard device coexists with a complete lack of strong local minimizers. The analysis hinges on the proof that every strong local minimizer in a hard device is also a global minimizer which is applicable much beyond the chosen class of models. Along the way we show that a new proof of sufficiency for a subclass of affine boundary conditions can be built around a novel nonlinear generalization of the classical Clapeyron theorem, whose subtle relation to dynamics was studied extensively by R. Fosdick.