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

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.