Variational Problems for Functionals Involving the Value Distribution

  • Giuseppe Buttazzo
  • Marc Oliver Rieger
Conference paper
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 68)

Abstract

We study variational problems involving the measure of level sets, or more precisely the push-forward of the Lebesgue measure. This problem generalizes variational problems with finitely many (discrete) volume constraints. We obtain existence results for this general framework. Moreover, we show the surprising existence of asymmetric solutions to symmetric variational problems with this type of volume constraints.

Keywords

Volume constraints level set constraints symmetric rearrangements minimization problems symmetry breaking solutions 
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References

  1. [1]
    E. Acerbi, I. Fonseca, and G. Mingione. Unpublished note.Google Scholar
  2. [2]
    N. Aguilera, H.W. Alt, and L.A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control Optim., 24 (1986), pp. 191–198.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), pp. 191–246.MATHMathSciNetGoogle Scholar
  4. [4]
    L. Ambrosio, I. Fonseca, P. Marcellini, and L. Tartar, On a volumeconstrained variational problem, Arch. Ration. Mech. Anal., 149 (1999), pp. 23–47.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.Google Scholar
  6. [6]
    G. Bouchitte and G. Buttazzo, New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal., 15 (1990), pp. 679–692.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    _____, Integral representation of nonconvex functionals defined on measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), pp. 101–117.MathSciNetMATHGoogle Scholar
  8. [8]
    _____, Relaxation for a class of nonconvex functionals defined on measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), pp. 345–361.MathSciNetMATHGoogle Scholar
  9. [9]
    G.R. Burton, Rearrangements of functions, maximization of convex functionals, and vortex rings, Mathematische Annalen, 276 (1987).Google Scholar
  10. [10]
    _____, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Metallurgica et Materialia, 163 (1989).Google Scholar
  11. [11]
    G.R. Burton and L. Preciso, Existence and isoperimetric characterisation of steady spherical vortex rings in a uniform flow in RN, Proceedings of the Royal Society of Edinburgh, (2004). to appear.Google Scholar
  12. [12]
    G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser., vol. 207, Longman, Harlow, 1989.Google Scholar
  13. [13]
    B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sciences, vol. 78, Springer Verlag, Berlin, 1989.Google Scholar
  14. [14]
    M.E. Gurtin, D. Polignone, and J. Vinals, Two-phase binary fluids and immissible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), pp. 815–831.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    B. Kawohl, Rearrangements and convexity of level sets in PDE, vol. 1150 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985.MATHGoogle Scholar
  16. [16]
    P. Laurence and E.W. Stredulinsky, Variational problems with topological constraints, Calculus of Variations and Partial Differential Equations, 10 (2000).Google Scholar
  17. [17]
    M. Morini and M.O. Rieger, On a volume constrained variational problem with lower order terms, Applied Mathematics and Optimization, 48 (2003), pp. 21–38.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    _____, Sharp existence results and the Γ-limit of variational problems with volume constraints. in preparation, 2004.Google Scholar
  19. [19]
    S. Mosconi and P. Tilli, Variational problems with several volume constraints on the level sets, Calc. Var. Partial Differential Equations, 14 (2002), pp. 233–247.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    S. Müller, Det = det. A remark on the distributional determinant, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), pp. 13–17.MATHGoogle Scholar
  21. [21]
    M.O. Rieger, Abstract variational problems with volume constraints, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004).Google Scholar
  22. [22]
    E. Stepanov and P. Tilli, On the Dirichlet problem with several volume constraints on the level sets, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 437–461.MathSciNetMATHGoogle Scholar
  23. [23]
    P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries, Interfaces Free Bound., 2 (2000), pp. 201–212.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Giuseppe Buttazzo
    • 1
  • Marc Oliver Rieger
    • 2
  1. 1.Dipartimento di MatematicaUniversitá di PisaPisaItaly
  2. 2.Department for MathematicsETH ZürichZürichSwitzerland

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