A free boundary problem arising in PDE optimization

  • Giuseppe ButtazzoEmail author
  • Edouard Oudet
  • Bozhidar Velichkov


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 



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).


  1. 1.
    Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge–Kantorovich equation. J. Eur. Math. Soc. 3, 139–168 (2001)zbMATHCrossRefGoogle Scholar
  2. 2.
    Brezis, H.: Multiplicateur de Lagrange en torsion “elasto-plastique”. Arch. Ration. Mech. Anal. 41, 32–40 (1971)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezis, H., Sibony, M.: Equivalence de deux inequations variationnelles et applications. Arch. Ration. Mech. Anal. 41, 254–265 (1971)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brezis, H., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. Fr. 96, 153–180 (1968)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bucur, D., Buttazzo, G.: Variational methods in shape optimization problems. In: Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser, Basel (2005)Google Scholar
  6. 6.
    Buttazzo, G., Carlier, G., Guarino, S.: Optimal regions for congested transport. Math. Model. Numer. Anal. (M2AN). (to appear)
  7. 7.
    Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Caffarelli, L.A., Friedman, A.: The free boundary for elastic–plastic torsion problem. Trans. Am. Math. Soc. 252, 65–97 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Caffarelli, L.A., Riviere, N.: On the Lipschitz character of the stress tensor when twisting an elastic–plastic bar. Arch. Ration. Mech. Anal. 69(1), 31–36 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Carlier, G., Jimenez, C., Santambrogio, F.: Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47(3), 1330–1350 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Clarke, F.H.: Multiple integrals of Lipschitz functions in the calculus of variations. Proc. Am. Math. Soc. 64(2), 260–264 (1977)zbMATHCrossRefGoogle Scholar
  12. 12.
    De Pascale, L., Evans, L.C., Pratelli, A.: Integral estimates for transport densities. Bull. Lond. Math. Soc. 36(3), 383–395 (2004)zbMATHCrossRefGoogle Scholar
  13. 13.
    De Pascale, L., Pratelli, A.: Regularity properties for Monge transport density and for solutions of some shape optimization problem. Calc. Var. PDE 14(3), 249–274 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
    De Philippis, G., Velichkov, B.: Existence and regularity of minimizers for some spectral optimization problems with perimeter constraint. Appl. Math. Optim. 69(2), 199–231 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Evans, L.C.: A second order elliptic equation with gradient constraint. Commun. PDE 4(5), 555–572 (1979)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1–22 (1990)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Ladyzhenskaya, O., Uraltseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)zbMATHGoogle Scholar
  18. 18.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. Theory Methods Appl. 12(11), 1203–1219 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Payne, L.E., Philippin, G.A.: Some applications of the maximum principle in the problem of the torsional creep. SIAM J. Appl. Math. 33(3), 446–455 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Santambrogio, F.: Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc. Var. 36, 343–354 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Santambrogio, F.: Private communicationGoogle Scholar
  22. 22.
    Ting, T.W.: Elastic–plastic torsion of square bar. Trans. Am. Math. Soc. 123, 369–401 (1966)zbMATHGoogle Scholar
  23. 23.
    Ting, T.W.: Elastic–plastic torsion problem II. Arch. Ration. Mech. Anal. 25, 342–365 (1967)zbMATHCrossRefGoogle Scholar
  24. 24.
    Ting, T.W.: Elastic–plastic torsion problem III. Arch. Ration. Mech. Anal. 34, 228–244 (1969)zbMATHCrossRefGoogle Scholar
  25. 25.
    Wiegner, M.: The \(C^{1,1}\)-character of solutions of second order elliptic equations with gradient constrained. Commun. Partial Differ. Equ. 6(3), 361–371 (1981)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Giuseppe Buttazzo
    • 1
    Email author
  • 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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