Geometric and Functional Analysis

, Volume 27, Issue 2, pp 373–426 | Cite as

Regularity of the optimal sets for some spectral functionals

  • Dario Mazzoleni
  • Susanna TerraciniEmail author
  • Bozhidar Velichkov


In this paper we study the regularity of the optimal sets for the shape optimization problem
$$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm open},\quad |\Omega| = 1\Big\},$$
where \({\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}\) denote the eigenvalues of the Dirichlet Laplacian and \({|\cdot|}\) the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer \({\Omega_{k}^{*}}\) is composed of a relatively open regular part which is locally a graph of a \({C^{\infty}}\) function and a closed singular part, which is empty if \({d < d^{*}}\), contains at most a finite number of isolated points if \({d = d^{*}}\) and has Hausdorff dimension smaller than \({(d-d^{*})}\) if \({d > d^{*}}\), where the natural number \({d^{*} \in [5,7]}\) is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.

Keywords and phrases

Shape optimization Dirichlet eigenvalues optimality conditions regularity of free boundaries viscosity solutions 

Mathematics Subject Classification

49Q10 (35R35, 47A75, 49R05) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ACS87.
    Aguilera N.E., Caffarelli L.A., Spruck J.: An optimization problem in heat conduction. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(3), 355–387 (1987)MathSciNetzbMATHGoogle Scholar
  2. AC81.
    Alt H.W., Caffarelli L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)MathSciNetzbMATHGoogle Scholar
  3. AFP00.
    L. Ambrosio, N. Fusco, and D. Pallara. Function of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).Google Scholar
  4. BGG69.
    Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  5. BL09.
    Briançon T., Lamboley J.: Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1149–1163 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Buc12.
    Bucur D.: Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch. Rational Mech. Anal. 206(3), 1073–1083 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. BB05.
    D. Bucur, G. Buttazzo. Variational methods in shape optimization problems. In: Progress in Nonlinear Differential Equations, Vol. 65. Birkhäuser, Basel (2005).Google Scholar
  8. BM15.
    Bucur D., Mazzoleni D.: A surgery result for the spectrum of the Dirichlet Laplacian. SIAM J. Math. Anal. 47(6), 4451–4466 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  9. BMPV15.
    Bucur D., Mazzoleni D., Pratelli A., Velichkov B.: Lipschitz regularity of the eigenfunctions on optimal domains. Arch. Ration. Mech. Anal. 216(1), 117–151 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  10. BM93.
    Buttazzo G., Dal Maso G.: An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122, 183–195 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Caf87.
    Caffarelli L.A.: A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are \({C^{1,\alpha}}\). Rev. Mat. Iberoamericana 3(2), 139–162 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Caf89.
    Caffarelli L.A.: A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz. Comm. Pure Appl. Math. 42(1), 55–78 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  13. CJK04.
    L.A. Caffarelli, D.S. Jerison, and C.E. Kenig. Global energy minimizers for free boundary problems and full regularity in three dimensions. Contemp. Math., Vol. 350. Amer. Math. Soc., Providence, RI (2004), pp. 83–97.Google Scholar
  14. CSY16.
    L.A. Caffarelli, H. Shahgholian, and K. Yeressian. A Minimization Problem with Free Boundary Related to a Cooperative System, preprint arXiv:1608.07689 (27 August 2016).
  15. Dav89.
    E. Davies. Heat Kernels and Spectral Theory. Cambridge University Press (1989).Google Scholar
  16. DV14.
    De Philippis G., Velichkov B.: Existence and regularity of minimizers for some spectral functionals with perimeter constraint. Appl. Math. Optim. 69, 199–231 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Sil11.
    De Silva D.: Free boundary regularity from a problem with right hand side. Interfaces and Free Boundaries 13(2), 223–238 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. DJ09.
    De Silva D., Jerison D.: A singular energy minimizing free boundary. J. Reine Angew. Math. 635, 1–21 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. DS15.
    De Silva D., Savin O.: A note on higher regularity boundary Harnack inequality. Discrete and Continuous Dynamical Systems Series A 35(12), 6155–6163 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  20. FH76.
    Friedland S., Hayman W. K.: Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helvetici 51, 133–161 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  21. Giu84.
    E. Giusti. Minimal Surfaces and Functions of Bounded Variations. Birkhäuser, Boston (1984).Google Scholar
  22. Hen06.
    A. Henrot. Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006).Google Scholar
  23. HP05.
    A. Henrot and M. Pierre. Variation et Optimisation de Formes. Une Analyse Géométrique. Mathématiques & Applications, Vol. 48, Springer, Berlin (2005).Google Scholar
  24. JK82.
    Jerison D.S., Kenig C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  25. JS15.
    Jerison D.S., Savin O.: Some remarks on stability of cones for the one phase free boundary problem. Geom. Funct. Anal. 25, 1240–1257 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. KT99.
    Kenig C.E., Toro T.: Free boundary regularity for harmonic measures and Poisson kernels. Ann. Math. 150, 369–454 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. KT97.
    Kenig C.E., Toro T.: Harmonic measure on locally flat domains. Duke Math. J. 87(3), 509–551 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  28. KN77.
    Kinderlehrer D., Nirenberg L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa 4(4), 373–391 (1977)MathSciNetzbMATHGoogle Scholar
  29. KL16.
    D. Kriventsov and F. Lin. Regularity for shape optimizers: the nondegenerate case, preprint arXiv:1609.02624 (9 September 2016).
  30. LY83.
    Li P., Yau S.-T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88(3), 309–318 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  31. LL97.
    E. Lieb and M. Loss. Analysis, Graduate studies in mathematics. AMS (1997)Google Scholar
  32. MP13.
    Mazzoleni D., Pratelli A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  33. RTT16.
    Ramos M., Tavares H., Terracini S.: Existence and regularity of solutions to optimal partition problems involving Laplacian eigenvalues. Arch. Rational Mech. Anal. 220, 363–443 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Sim83.
    L. Simon. Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis. Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Camberra (1983).Google Scholar
  35. Sim68.
    Simons J.: Minimal varieties in riemannian manifolds. Ann. of Math. 88(2), 62– (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  36. Spe73.
    Sperner E.: Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134, 317–327 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  37. TT12.
    Tavares H., Terracini S.: Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. PDE 45, 273–317 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  38. Wei99.
    Weiss G. S.: Partial regularity for a minimum problem with free boundary. J. Geom. Anal. 9(2), 317–326 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Dario Mazzoleni
    • 1
  • Susanna Terracini
    • 1
    Email author
  • Bozhidar Velichkov
    • 2
  1. 1.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTurinItaly
  2. 2.Laboratoire Jean Kuntzmann (LJK)Université Grenoble Alpes Bâtiment IMAGSaint-Martin-d’HèresFrance

Personalised recommendations