On functions whose symmetric part of gradient agree and a generalization of Reshetnyak's compactness theorem

TL;DR

This paper investigates when two mappings with equal symmetric parts of their gradients are related by a rotation, extending classical results like Liouville's theorem and Reshetnyak's theorem to less regular functions and sequences.

Contribution

It generalizes Reshetnyak's compactness theorem to pairs of weakly converging sequences with bounded dilatation, establishing conditions under which their gradients differ by a rotation.

Findings

Proves the equivalence of gradients up to rotation under specified integrability conditions.

Extends classical theorems to mappings with less regularity and sequences with bounded dilatation.

Identifies sharp conditions in two dimensions for the main result.

Abstract

We consider the following question: Given a connected open domain $\Omega\subset R^n$, suppose $u,v:\Omega\rightarrow R^n$ with $\det(\nabla u)>0$, $\det(\nabla v)>0$ a.e. are such that $\nabla u^T(x)\nabla u(x)=\nabla v(x)^T \nabla v(x)$ a.e. does this imply a global relation of the form $\nabla v(x)= R\nabla u(x)$ a.e. in $\Omega$ where $R\in SO(n)$? If $u,v$ are $C^1$ it is an exercise to see this true, if $u,v\in W^{1,1}$ we show this is false. We prove this question has a positive answer if $v\in W^{1,1}$ and $u\in W^{1,n}$ is a mapping of $L^p$ integrable dilatation for $p>n-1$. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville's theorem that states that the differential inclusion $\nabla u\in SO(n)$ can only be satisfied by an affine mapping. Liouville's corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence $v_k\in W^{1,1}$ for which $$ \int_{\Omega} \mathrm{dist}(\nabla v_k,SO(n)) dz\rightarrow 0 \text{as} k\rightarrow \infty. $$ Let $S(\cdot)$ denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous for any pair of weakly converging sequences $v_k\in W^{1,p}$ and $u_k\in W^{1,\frac{p(n-1)}{p-1}}$ (where $p\in \left[1,n\right]$ and the sequence $(u_k)$ has its dilatation pointwise bounded above by an $L^r$ integrable function, $r>n-1$) that satisfy $\int_{\Omega} \left|S(\nabla u_k)-S(\nabla v_k)\right|^p dz\rightarrow 0$ as $k\rightarrow \infty$ and for which the sign of the $\det(\nabla v_k)$ tends to 1 in $L^1$. This result contains Reshetnyak's theorem as the special case $(u_k)\equiv Id$, p=1.