A free boundary problem arising in PDE optimization


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.

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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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Correspondence to Giuseppe Buttazzo.

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Communicated by L. Ambrosio.

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Buttazzo, G., Oudet, E. & Velichkov, B. A free boundary problem arising in PDE optimization. Calc. Var. 54, 3829–3856 (2015). https://doi.org/10.1007/s00526-015-0923-1

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Mathematics Subject Classification

  • 49J45
  • 35R35
  • 49M05
  • 35J25