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A free boundary problem arising in PDE optimization

  • Giuseppe Buttazzo
  • Edouard Oudet
  • Bozhidar Velichkov
Article

Abstract

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form
$$\begin{aligned} \sup _{\int _D\theta \,dx=m}\ \inf _{u\in H^1_0(D)}\int _D\left( \frac{1+\theta }{2}|\nabla u|^2-fu\right) dx. \end{aligned}$$
We prove the existence of an optimal reinforcement \(\theta \) and that it has some higher integrability properties. We also provide some numerical computations for \(\theta \) and u.

Mathematics Subject Classification

49J45 35R35 49M05 35J25 

Notes

Acknowledgments

A part of this paper was written during a visit of the authors at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of Linz. The authors gratefully acknowledge the Institute for the excellent working atmosphere provided. The work of the first author is part of the Project 2010A2TFX2 “Calcolo delle Variazioni” funded by the Italian Ministry of Research and University. The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Giuseppe Buttazzo
    • 1
  • Edouard Oudet
    • 2
  • Bozhidar Velichkov
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Laboratoire Jean Kuntzmann (LJK)Université Joseph FourierGrenoble Cedex 9France

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