Abstract
We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of k mutually disjoint open sets which have a \({\mathcal {H}}^{d-1}\)-countably rectifiable boundary and are contained into a given box D in \(\mathbb {R}^d\).
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References
Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)
Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111(4), 291–322 (1990)
Ambrosio, N., Fusco, L., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)
Bogosel, B., Velichkov, B.: A multiphase shape optimization problem for eigenvalues: qualitative study and numerical results. SIAM J. Numer. Anal. 54(1), 210–241 (2016)
Bonnaillie-Noël, V., Helffer, B., Vial, G.: Numerical simulations for nodal domains and spectral minimal partitions. ESAIM Control Optim. Calc. Var. 16(1), 221–246 (2010)
Bucur, D., Buttazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8(2), 571–579 (1998)
Bucur, D., Fragalà, I.: On the honeycomb conjecture for Robin Laplacian eigenvalues. ArXiv e-prints (2017)
Bucur, D., Fragalà, I.: Proof of the honeycomb asymptotics for optimal Cheeger clusters. ArXiv e-prints (2017)
Bucur, D., Fragalà, I., Velichkov, B., Verzini, G.: On the honeycomb conjecture for a class of minimal convex partitions. Trans. AMS. https://doi.org/10.1090/tran/7326 (2017) (to appear)
Bucur, D., Giacomini, A.: A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal. 198(3), 927–961 (2010)
Bucur, D., Giacomini, A.: Faber–Krahn inequalities for the Robin–Laplacian: a free discontinuity approach. Arch. Ration. Mech. Anal. 218(2), 757–824 (2015)
Bucur, D., Giacomini, A.: Shape optimization problems with Robin conditions on the free boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(6), 1539–1568 (2016)
Bucur, D., Luckhaus, S.: Monotonicity formula and regularity for general free discontinuity problems. Arch. Ration. Mech. Anal. 211(2), 489–511 (2014)
Bucur, D., Velichkov, B.: Multiphase shape optimization problems. SIAM J. Control Optim. 52(6), 3556–3591 (2014)
Caffarelli, L.A., Lin, F.H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1–2), 5–18 (2007)
Caffarelli, L.A., Kriventsov, D.: A free boundary problem related to thermal insulation. Commun. Partial Differ. Equ. 41(7), 1149–1182 (2016)
Conti, M., Terracini, S., Verzini, G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198(1), 160–196 (2003)
Conti, M., Terracini, S., Verzini, G.: On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae. Calc. Var. Partial Differ. Equ. 22(1), 45–72 (2005)
Conti, M., Terracini, S., Verzini, G.: A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54(3), 779–815 (2005)
De Giorgi, E., Ambrosio, L.: New functionals in the calculus of variations. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), no. 2, 199–210 (1989)
De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108(3), 195–218 (1989)
Helffer, B.: Domaines nodaux et partitions spectrales minimales (d’après B. Helffer, T. Hoffmann-Ostenhof et S. Terracini), Séminaire: Équations aux Dérivées Partielles. 2006–2007, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, pp. Exp. No. VIII, 23 (2007)
Helffer, B.: On spectral minimal partitions: a survey. Milan J. Math. 78(2), 575–590 (2010)
Helffer, B., Hoffmann-Ostenhof, T., Terracini, S.: Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 101–138 (2009)
Kriventsov, D.: A free boundary problem related to thermal insulation: flat implies smooth. ArXiv e-prints (2015)
Leonardi, G.P.: An overview on the Cheeger problem. In: Pratelli, A., Leugering, G. (eds.) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol. 166, pp. 117–139. Springer, Bern (2016)
Ramos, M., Tavares, H., Terracini, S.: Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues. Arch. Ration. Mech. Anal. 220, 363–443 (2016)
Acknowledgements
The first author was supported by the “Geometry and Spectral Optimization” research programme LabEx PERSYVAL-Lab GeoSpec (ANR-11-LABX-0025-01) and ANR Comedic (ANR-15-CE40-0006).
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Communicated by L. Ambrosio.
DB is member of the Institut Universitaire de France. IF and AG are members of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Bucur, D., Fragalà, I. & Giacomini, A. Optimal partitions for Robin Laplacian eigenvalues. Calc. Var. 57, 122 (2018). https://doi.org/10.1007/s00526-018-1393-z
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DOI: https://doi.org/10.1007/s00526-018-1393-z