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

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

2011-05-19

arxiv.org/abs/1105.3993v2

Mathematics

Analysis of PDEs

35 pages

Scientific paper

We consider the following question: Given a connected open domain $\Omega\subset R^n$, suppose $u,v:\Omega\rightarrow R^n$ with $\det(\na u)>0$, $\det(\na v)>0$ a.e.\ are such that $\na u^T(x)\na u(x)=\na v(x)^T \na v(x)$ a.e.\, does this imply a global relation of the form $\na v(x)= R\na u(x)$ 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 $\na u\in SO(n)$ 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 $\int_{\Omega} \mathrm{dist}(\na v_k,SO(n)) dz\rightarrow 0\text{as}k\rightarrow \infty$. We prove an analogous result for any pair of weakly converging sequences $v_k\in W^{1,1}$ and $u_k\in W^{1,n}$ that satisfy certain necessary conditions and whose symmetric parts of gradient become increasingly close in $L^1$ norm. This result contains Reshetnyak's theorem as the special case $(u_k)\equiv Id$.

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