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Large m asymptotics for minimal partitions of the Dirichlet eigenvalue

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Abstract

In this paper, we study large m asymptotics of the l1 minimal m-partition problem for the Dirichlet eigenvalue. For any smooth domain Ω ⊂ ℝn such that ∣Ω∣ = 1, we prove that the limit \({\rm{lim}}_{m \to \infty}l_m^1\left(\Omega \right) = {c_0}\) exists, and the constant c0 is independent of the shape of Ω. Here, \(l_m^1\left({\rm{\Omega}} \right)\) denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition of Ω.

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References

  1. Alper O. On the singular set of free interface in an optimal partition problem. Comm Pure Appl Math, 2020, 73: 855–915

    Article  MathSciNet  Google Scholar 

  2. Bonnaillie-Noël V, Helffer B. Nodal and spectral minimal partitions—The state of the art in 2016. In: Shape Optimization and Spectral Theory. Warsaw: De Gruyter Open, 2017, 353–397

    Chapter  Google Scholar 

  3. Bourgain J. On Pleijel’s nodal domain theorem. Int Math Res Not IMRN, 2015, 2015: 1601–1612

    Article  MathSciNet  Google Scholar 

  4. Bucur D. Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch Ration Mech Anal, 2012, 206: 1073–1083

    Article  MathSciNet  Google Scholar 

  5. Bucur D, Buttazzo G, Henrot A. Existence results for some optimal partition problems. Adv Math Sci Appl, 1998, 8: 571–579

    MathSciNet  MATH  Google Scholar 

  6. Bucur D, Fragala I, Velichkov B, et al. On the honeycomb conjecture for a class of minimal convex partitions. Trans Amer Math Soc, 2018, 370: 7149–7179

    Article  MathSciNet  Google Scholar 

  7. Bucur D, Zolesio J P. W-dimensional shape optimization under capacitary constraint. J Differential Equations, 1995, 123: 504–522

    Article  MathSciNet  Google Scholar 

  8. Buttazzo G, Dal Maso G. Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl Math Optim, 1991, 23: 17–49

    Article  MathSciNet  Google Scholar 

  9. Caffarelli L A, Lin F H. An optimal partition problem for eigenvalues. J Sci Comput, 2007, 31: 5–18

    Article  MathSciNet  Google Scholar 

  10. Caffarelli L A, Lin F H. Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J Amer Math Soc, 2008, 21: 847–862

    Article  MathSciNet  Google Scholar 

  11. Helffer B. On spectral minimal partitions: A survey. Milan J Math, 2010, 78: 575–590

    Article  MathSciNet  Google Scholar 

  12. Lin F H. Extremum problems of Laplacian eigenvalues and generalized Polya conjecture. Chin Ann Math Ser B, 2017, 38: 497–512

    Article  MathSciNet  Google Scholar 

  13. Mazzoleni D, Pratelli A. Existence of minimizers for spectral problems. J Math Pures Appl (9), 2013, 100: 433–453

    Article  MathSciNet  Google Scholar 

  14. Steinerberger S. A geometric uncertainty principle with an application to Pleijel’s estimate. Ann Henri Poincare, 2014, 15: 2299–2319

    Article  MathSciNet  Google Scholar 

  15. Sverak V. On optimal shape design. J Math Pures Appl (9), 1993, 72: 537–551

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Science Foundation of USA (Grant Nos. DMS-1501000 and DMS-1955249).

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  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Zhiyuan Geng & Fanghua Lin

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  1. Zhiyuan Geng
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  2. Fanghua Lin
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Correspondence to Fanghua Lin.

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Geng, Z., Lin, F. Large m asymptotics for minimal partitions of the Dirichlet eigenvalue. Sci. China Math. 65, 1–8 (2022). https://doi.org/10.1007/s11425-020-1802-6

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  • DOI: https://doi.org/10.1007/s11425-020-1802-6

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