# 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

, Volume 56 (4) – Jun 6, 2017
31 pages

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/lp/springer_journal/variational-methods-for-the-selection-of-solutions-to-an-implicit-OoAgo0Aqv2
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1185-x
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jun 6, 2017

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