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We consider the vectorial system $$\begin{aligned} {\left\{ \begin{array}{ll} Du \in \mathcal {O}(2), &{} \text{ a.e. } \text{ in }\,\;\Omega , \\ u=0, &{} \text{ on } \,\;\partial \Omega , \end{array}\right. } \end{aligned}$$ D u ∈ O ( 2 ) , a.e. in Ω , u = 0 , on ∂ Ω , where $$\Omega $$ Ω is a subset of $$\mathbb R^2$$ R 2 , $$u:\Omega \rightarrow \mathbb R^2$$ u : Ω → R 2 and $$\mathcal {O}(2)$$ O ( 2 ) is the orthogonal group of $$\mathbb R^2$$ R 2 . We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appropriate weighted measure of some set of singularities of the gradient.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Jun 6, 2017
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