On multiwell Liouville theorems in higher dimension

TL;DR

This paper establishes new Liouville theorems for certain matrix subsets in higher dimensions, showing under specific conditions that functions close to these sets are approximated by rigid motions, extending known results from two to higher dimensions.

Contribution

The paper proves higher-dimensional Liouville theorems for matrix sets with specific connectivity, generalizing known two-dimensional results to dimensions three and above.

Findings

Results hold for m=2 and generic m≤n in n≥3 dimensions.

Provides quantitative estimates linking the distance of derivatives to matrix sets.

Extends classical 2D Liouville theorems to higher dimensions under new conditions.

Abstract

We consider certain subsets of the space of $n\times n$ matrices of the form $K = \cup_{i=1}^m SO(n)A_i$, and we prove that for $p>1, q \geq 1$ and for connected $\Omega'\subset\subset\Omega\subset \R^n$, there exists positive constant $a<1$ depending on $n,p,q, \Omega, \Omega'$ such that for $ \veps=\| {dist}(Du, K)\|_{L^p(\Omega)}^p$ we have $\inf_{R\in K}\|Du-R\|^p_{L^p(\Omega')}\leq M\veps^{1/p}$ provided $u$ satisfies the inequality $\| D^2 u\|_{L^q(\Omega)}^q\leq a\veps^{1-q}$. Our main result holds whenever $m=2$, and also for {\em generic} $m\le n$ in every dimension $n\ge 3$, as long as the wells $SO(n)A_1,..., SO(n)A_m$ satisfy a certain connectivity condition. These conclusions are mostly known when $n=2$, and they are new for $n\ge 3$.