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Minimization of the k-th eigenvalue of the Dirichlet Laplacian

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Abstract

For every \({k \in \mathbb{N}}\), we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.

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References

  1. 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 

  2. Briançon T., Hayouni M., Pierre M.: Lipschitz continuity of state functions in some optimal shaping. Calc. Var. Partial Differ. Equ. 23, 13–32 (2005)

    Article  MATH  Google Scholar 

  3. Bucur D.: Uniform concentration-compactness for Sobolev spaces on variable domains. J. Differ. Equ. 162, 427–450 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bucur D.: Regularity of optimal convex shapes. J. Convex Anal. 10, 501–516 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Bucur, D., Buttazzo, G.: Variational methods in shape optimization problems. In: Progress in Nonlinear Differential Equations and their Applications, vol. 65, Birkhäuser, Boston, 2005

  6. Bucur D., Henrot A.: Minimization of the third eigenvalue of the Dirichlet Laplacian. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, 985–996 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. 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  ADS  MATH  Google Scholar 

  8. Caffarelli, L., Salsa, S.: A geometric approach to free boundary problems. In: Graduate Studies in Mathematics. American Mathematical Society, Providence, 2005

  9. Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge, 1990

  10. Dunford, N., Schwartz, J.T.: Linear operators. Part II: Spectral theory. In: Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle. Wiley, New York, 1963

  11. Hayouni M., Pierre M.: Domain continuity for an elliptic operator of fourth order. Commun. Contemp. Math. 4, 1–14 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Henrot, A.: Extremum problems for eigenvalues of elliptic operators. In: Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006

  13. Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. CVGMT (2011) (Preprint)

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  1. Laboratoire de Mathématiques, CNRS UMR 5127, Université de Savoie, Campus Scientifique, 73376, Le-Bourget-Du-Lac, France

    Dorin Bucur

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  1. Dorin Bucur
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Correspondence to Dorin Bucur.

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Communicated by G. Dal Maso

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Bucur, D. Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch Rational Mech Anal 206, 1073–1083 (2012). https://doi.org/10.1007/s00205-012-0561-0

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  • DOI: https://doi.org/10.1007/s00205-012-0561-0

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