Ricerche di Matematica

, Volume 68, Issue 2, pp 359–373 | Cite as

Density lower bound estimate for local minimizer of free interface problem with volume constraint

  • Luca EspositoEmail author


We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).


Free boundary problem Perimeter penalization Volume constraint Regularity 

Mathematics Subject Classification

49N60 49Q20 


  1. 1.
    Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 107–144 (1981)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Buttazzo, G.: An optimal design problem with perimeter penalization. Calc. Var. Part. Differ. Equ. 1, 55–69 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  4. 4.
    Bombieri, E.: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 78, 99–130 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carozza, M., Fonseca, I., Di Napoli, A.Passarelli: Regularity results for an optimal design problem with a volume constraint, ESAIM Control. Optim. Calc. Var. 20(2), 460–487 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    David, G.: \(C^1\) —arcs for minimizers of the Mumford–Shah functional. SIAM J. Appl. Math. 56, 783–888 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    De Philippis, G., Figalli, A.: A note on the dimension of the singular set in free interface problems. Differ. Integral Equ. 28(5/6), 523–536 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Esposito, L., Fusco, N.: A remark on a free interface problem with volume constraint. J. Convex Anal. 18(2), 417–426 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fonseca, I., Fusco, N.: Regularity results for anisotropic image segmentation models. Ann. Sc. Norm. Super. Pisa 24, 463–499 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal. 186, 477–537 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fusco, N., Julin, V.: On the regularity of critical and minimal sets of a free interface problem. arXiv:1309.6810
  12. 12.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, River Edge, NJ (2003)CrossRefGoogle Scholar
  13. 13.
    Gurtin, M.: On phase transitions with bulk, interfacial, and boundary enwergy. Arch. Ration. Mech. Anal. 96, 243–264 (1986)CrossRefGoogle Scholar
  14. 14.
    Kristensen, J., Mingione, G.: The singular set of lipschitzian minima of multiple integrals. Arch. Ration. Mech. Anal. 184, 341–369 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Larsen, C.J.: Regularity of componrnts in optimal design problems with perimeter penalization. Calc. Var. Part. Differ. Equ. 16, 17–29 (2003)CrossRefGoogle Scholar
  16. 16.
    Li, H., Halsey, T., Lobkovsky, A.: Singular shape of a fluid drop in an electric or magnetic field. Europhys. Lett. 27, 575–580 (1994)CrossRefGoogle Scholar
  17. 17.
    Lin, F.H.: Variational problems with free interfaces. Calc. Var. Part. Differ. Equ. 1, 149–168 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin, F.H., Kohn, R.V.: Partial regularity for optimal design problems involving both bulk and surface energies. Chin. Ann. Math. 20B(2), 137–158 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maddalena, F., Solimini, S.: Regularity properties of free discontinuity sets. Ann. Inst. H. Poincaré Anal. Non Lin éaire 18, 675–685 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge Univeristy Press, Cambridge (2012)CrossRefGoogle Scholar
  21. 21.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the centre for mathematical analysis, Australian National University, centre for mathematical analysis, vol. 3, Canberra (1983)Google Scholar
  22. 22.
    Ŝverák, V., Yan, X.: Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99, 15269–15276 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tamanini, I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Taylor, G.I.: Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397 (1964)CrossRefGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaFiscianoItaly

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