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Regularity of all minimizers of a class of spectral partition problems

  • Received: 10 February 2020 Accepted: 08 July 2020 Published: 20 July 2020
  • We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs $ (\omega_1, \dots, \omega_m) \mapsto \sum\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod\limits_{i = 1}^{m} \left( \prod\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right) $ where $(\omega_1, \dots, \omega_m)$ are the sets of the partition and $\lambda_{j}(\omega_i)$ is the $j$-th Laplace eigenvalue of the set $\omega_i$ with zero Dirichlet boundary conditions.

    Citation: Hugo Tavares, Alessandro Zilio. Regularity of all minimizers of a class of spectral partition problems[J]. Mathematics in Engineering, 2021, 3(1): 1-31. doi: 10.3934/mine.2021002

    Related Papers:

  • We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs $ (\omega_1, \dots, \omega_m) \mapsto \sum\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)^{p_i}\right)^{1/p_i}, \quad \prod\limits_{i = 1}^{m} \left( \prod\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right), \quad \prod\limits_{i = 1}^{m} \left( \sum\limits_{j = 1}^{k_i} \lambda_{j}(\omega_i)\right) $ where $(\omega_1, \dots, \omega_m)$ are the sets of the partition and $\lambda_{j}(\omega_i)$ is the $j$-th Laplace eigenvalue of the set $\omega_i$ with zero Dirichlet boundary conditions.


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    [1] Alper O (2020) On the singular set of free interfacein an optimal partition problem. Commun Pure Appl Math 73: 855-915.
    [2] Alt H, Caffarelli L, Friedman A (1984) Variational problems with two phases and their free boundaries. T Am Math Soc 282: 431-461.
    [3] Band R, Berkolaiko G, Raz H, et al.(2012) The number of nodal domains on quantum graphs as a stability index of graph partitions. Commun Math Phys 311: 815-838.
    [4] Berkolaiko G, Kuchment P, Smilansky U (2012) Critical partitions and nodal deficiency of billiard eigenfunctions. Geom Funct Anal 22: 1517-1540.
    [5] Bonnaillie-Noël V, Helffer B, Vial G (2010) Numerical simulations for nodal domains and spectral minimal partitions. ESAIM Contr Optim Ca 16: 221-246.
    [6] Bourdin B, Bucur D, Oudet E (2009) Optimal partitions for eigenvalues. SIAM J Sci Comput 31: 4100-4114.
    [7] Bucur D, Buttazzo G (2005) Variational Methods in Shape Optimization Problems, Boston: Birkhäuser Boston Inc.
    [8] Bucur D, Buttazzo G, Henrot A (1998) Existence results for some optimal partition problems. Adv Math Sci Appl 8: 571-579.
    [9] Caffarelli L, Lin FH (2007) An optimal partition problem for eigenvalues. J Sci Comput 31: 5-18.
    [10] Caffarelli L, Lin FH (2010) Analysis on the junctions of domain walls. Discrete Contin Dyn Syst 28: 915-929.
    [11] Chang SM, Lin CS, Lin TC, et al (2004) Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys D 196: 341-361.
    [12] Conti M, Terracini S, Verzini G (2002) Nehari's problem and competing species systems. Ann I H Poincaré Anal Non Linéaire 19: 871-888.
    [13] Conti M, Terracini S, Verzini G (2003) An optimal partition problem related to nonlinear eigenvalues. J Funct Anal 198: 160-196.
    [14] Conti M, Terracini S, Verzini G (2005) On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae. Calc Var Partial Dif 22: 45-72.
    [15] De Philippis G, Lamboley G, Pierre M, et al.(2018) Regularity of minimizers of shape optimization problems involving perimeter. J Math Pure Appl 109: 147-181.
    [16] Hecht F (2012) New development in freefem++. J Numer Math 20: 251-265.
    [17] Helffer B, Hoffmann-Ostenhof T, Terracini S (2009) Nodal domains and spectral minimal partitions. Ann I H Poincaré Anal Non Linéaire 26: 101-138.
    [18] Helffer B, Hoffmann-Ostenhof T, Terracini S (2010) Nodal minimal partitions in dimension 3. Discrete Contin Dyn Syst 28: 617-635.
    [19] Noris B, Tavares H, Terracini S, et al. (2010) Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun Pure Appl Math 63: 267-302.
    [20] Ramos M, Tavares H, Terracini S (2016) Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues. Arch Ration Mech Anal 220: 363-443.
    [21] Soave N, Tavares H, Terracini S, et al. (2016) Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping. Nonlinear Anal 138: 388-427.
    [22] Soave N, Terracini S (2015) Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modelling phase separation. Adv Math 279: 29-66.
    [23] Tavares H, Terracini S (2012) Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc Var Partial Dif 45: 273-317.
    [24] Tavares H, Terracini S (2012) Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems. Ann I H Poincaré Anal Non Linéaire 29: 279-300.
    [25] Terracini S, Verzini G, Zilio A (2016) Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian. J Eur Math Soc 18: 2865-2924.
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