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

created by lorent on 20 May 2011
modified on 18 Feb 2014

[BibTeX]

Published Paper

Inserted: 20 may 2011
Last Updated: 18 feb 2014

Journal: Calculus of Variations and PDE
Volume: 48
Number: 3-4
Pages: 625-665
Year: 2013

Abstract:

We consider the following question: Given a connected open domain $\Omega\subset R^n$, suppose $u,v:\Omega\rightarrow R^n$ with $\det(Du)>0$, $\det(Dv)>0$ a.e.\ are such that $Du^T(x)Du(x)=Dv(x)^T Dv(x)$ a.e. , does this imply a global relation of the form $Dv(x)= R Du(x)$ 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. In Theorem 3 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 $Du\in SO(n)$ 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 result 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 $p\in [1,n]$ and the sequence $(u_k)$ has its dilatation pointwise bounded above by an $L^r$ integrable function, $r>n-1$) that satisfy $\int_{\Omega} |S(\nabla u_k)-S(\nabla v_k)|^p dz\rightarrow 0$ 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 $(u_k)\equiv Id$, $p=1$.