A free boundary problem arising in PDE optimization

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge–Kantorovich equation. J. Eur. Math. Soc. 3, 139–168 (2001)

    MATH  Article  Google Scholar 

  2. 2.

    Brezis, H.: Multiplicateur de Lagrange en torsion “elasto-plastique”. Arch. Ration. Mech. Anal. 41, 32–40 (1971)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brezis, H., Sibony, M.: Equivalence de deux inequations variationnelles et applications. Arch. Ration. Mech. Anal. 41, 254–265 (1971)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Brezis, H., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. Fr. 96, 153–180 (1968)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Bucur, D., Buttazzo, G.: Variational methods in shape optimization problems. In: Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser, Basel (2005)

  6. 6.

    Buttazzo, G., Carlier, G., Guarino, S.: Optimal regions for congested transport. Math. Model. Numer. Anal. (M2AN). http://cvgmt.sns.it (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)

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Caffarelli, L.A., Friedman, A.: The free boundary for elastic–plastic torsion problem. Trans. Am. Math. Soc. 252, 65–97 (1979)

    MATH  MathSciNet  Article  Google 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)

    MATH  MathSciNet  Article  Google 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)

    MATH  MathSciNet  Article  Google 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)

    MATH  Article  Google 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)

    MATH  Article  Google 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)

    MATH  Article  Google 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)

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Evans, L.C.: A second order elliptic equation with gradient constraint. Commun. PDE 4(5), 555–572 (1979)

    MATH  Article  Google Scholar 

  16. 16.

    Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1–22 (1990)

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Ladyzhenskaya, O., Uraltseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    Google Scholar 

  18. 18.

    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. Theory Methods Appl. 12(11), 1203–1219 (1988)

    MATH  MathSciNet  Article  Google 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)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Santambrogio, F.: Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc. Var. 36, 343–354 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    Santambrogio, F.: Private communication

  22. 22.

    Ting, T.W.: Elastic–plastic torsion of square bar. Trans. Am. Math. Soc. 123, 369–401 (1966)

    MATH  Google Scholar 

  23. 23.

    Ting, T.W.: Elastic–plastic torsion problem II. Arch. Ration. Mech. Anal. 25, 342–365 (1967)

    MATH  Article  Google Scholar 

  24. 24.

    Ting, T.W.: Elastic–plastic torsion problem III. Arch. Ration. Mech. Anal. 34, 228–244 (1969)

    MATH  Article  Google 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)

    MATH  MathSciNet  Article  Google Scholar 

Download references

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Giuseppe Buttazzo.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Mathematics Subject Classification

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