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

  • Gisella Croce
  • Giovanni PisanteEmail author


We consider the vectorial system
$$\begin{aligned} {\left\{ \begin{array}{ll} Du \in \mathcal {O}(2), &{} \text{ a.e. } \text{ in }\,\;\Omega , \\ u=0, &{} \text{ on } \,\;\partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a subset of \(\mathbb R^2\), \(u:\Omega \rightarrow \mathbb R^2\) and \(\mathcal {O}(2)\) is the orthogonal group of \(\mathbb R^2\). 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.

Mathematics Subject Classification

34A60 35A15 35F30 49J40 49Q15 



G.P. is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors wish to thank the anonymous referees that carefully read the manuscript and whose suggestions have been extremely valuable for the final version of the work.


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Authors and Affiliations

  1. 1.Normandie UnivUNIHAVRE, LMAH, FR-CNRS-3335Le HavreFrance
  2. 2.Dipartimento di Matematica e FisicaUniversità degli Studi della Campania “Luigi Vanvitelli”CasertaItaly

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