Multiwell rigidity in nonlinear elasticity

TL;DR

This paper establishes a quantitative rigidity estimate for multiwell problems in nonlinear elasticity, showing that gradient fields close to multiple wells are actually close to a single well under certain conditions.

Contribution

It provides the first optimal-scale rigidity estimate for multiwell nonlinear elasticity problems, linking proximity to multiple wells with proximity to a single well.

Findings

Proves a quantitative rigidity estimate for multiwell nonlinear elasticity.

Shows that gradient fields near multiple wells are close to one well if interface length is bounded.

The estimate holds for any connected subdomain with optimal scaling.

Abstract

We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity. Precisely, we show that if a gradient field is L^1-close to a set of the form SO(n)U_1 \cup ... \cup SO(n)U_l, and an appropriate bound on the length of the interfaces holds, then the gradient field is actually close to only one of the wells SO(n)U_i. The estimate holds for any connected subdomain, and has the optimal scaling.