Abstract
We consider the following question: Given a connected open domain \({\Omega \subset \mathbb{R}^n}\), suppose \({u, v : \Omega \rightarrow \mathbb{R}^n}\) 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 Ω 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 1 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
Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 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 → ∞ 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 ) ≡ Id, p = 1.
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Lorent, A. On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem. Calc. Var. 48, 625–665 (2013). https://doi.org/10.1007/s00526-012-0566-4
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DOI: https://doi.org/10.1007/s00526-012-0566-4