Variational methods for the selection of solutions to an implicit system of PDE

Variational methods for the selection of solutions to an implicit system of PDE 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

Variational methods for the selection of solutions to an implicit system of PDE

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Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
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