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Regularity of the optimal sets for some spectral functionals

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Abstract

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.

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References

  1. 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)

    MathSciNet  MATH  Google Scholar 

  2. Alt H.W., Caffarelli L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)

    MathSciNet  MATH  Google Scholar 

  3. L. Ambrosio, N. Fusco, and D. Pallara. Function of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).

  4. Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bucur D.: Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch. Rational Mech. Anal. 206(3), 1073–1083 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

  8. Bucur D., Mazzoleni D.: A surgery result for the spectrum of the Dirichlet Laplacian. SIAM J. Math. Anal. 47(6), 4451–4466 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buttazzo G., Dal Maso G.: An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122, 183–195 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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.

  14. 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. E. Davies. Heat Kernels and Spectral Theory. Cambridge University Press (1989).

  16. De Philippis G., Velichkov B.: Existence and regularity of minimizers for some spectral functionals with perimeter constraint. Appl. Math. Optim. 69, 199–231 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Silva D.: Free boundary regularity from a problem with right hand side. Interfaces and Free Boundaries 13(2), 223–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. De Silva D., Jerison D.: A singular energy minimizing free boundary. J. Reine Angew. Math. 635, 1–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Giusti. Minimal Surfaces and Functions of Bounded Variations. Birkhäuser, Boston (1984).

  22. A. Henrot. Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006).

  23. A. Henrot and M. Pierre. Variation et Optimisation de Formes. Une Analyse Géométrique. Mathématiques & Applications, Vol. 48, Springer, Berlin (2005).

  24. Jerison D.S., Kenig C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kenig C.E., Toro T.: Free boundary regularity for harmonic measures and Poisson kernels. Ann. Math. 150, 369–454 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kenig C.E., Toro T.: Harmonic measure on locally flat domains. Duke Math. J. 87(3), 509–551 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kinderlehrer D., Nirenberg L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa 4(4), 373–391 (1977)

    MathSciNet  MATH  Google Scholar 

  29. D. Kriventsov and F. Lin. Regularity for shape optimizers: the nondegenerate case, preprint arXiv:1609.02624 (9 September 2016).

  30. Li P., Yau S.-T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88(3), 309–318 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Lieb and M. Loss. Analysis, Graduate studies in mathematics. AMS (1997)

  32. Mazzoleni D., Pratelli A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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).

  35. Simons J.: Minimal varieties in riemannian manifolds. Ann. of Math. 88(2), 62– (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sperner E.: Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134, 317–327 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Weiss G. S.: Partial regularity for a minimum problem with free boundary. J. Geom. Anal. 9(2), 317–326 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto, 10, 10123, Turin, Italy

    Dario Mazzoleni & Susanna Terracini

  2. Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes Bâtiment IMAG, 700 Avenue Centrale, 38401, Saint-Martin-d’Hères, France

    Bozhidar Velichkov

Authors
  1. Dario Mazzoleni
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  2. Susanna Terracini
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  3. Bozhidar Velichkov
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Correspondence to Susanna Terracini.

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Mazzoleni, D., Terracini, S. & Velichkov, B. Regularity of the optimal sets for some spectral functionals. Geom. Funct. Anal. 27, 373–426 (2017). https://doi.org/10.1007/s00039-017-0402-2

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  • DOI: https://doi.org/10.1007/s00039-017-0402-2

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