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A Faber–Krahn Inequality for the Cheeger Constant of \(N\)-gons

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Abstract

We prove that the regular \(N\)-gon minimizes the Cheeger constant among polygons with a given area and \(N\) sides.

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References

  1. Alter, F., Caselles, V.: Uniqueness of the Cheeger set of a convex body. Nonlinear Anal. 70(1), 32–44 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchitté, G., Fragalà, I., Lucardesi, I., Seppecher, P.: Optimal thin torsion rods and Cheeger sets. SIAM J. Math. Anal. 44(1), 483–512 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brooks, R., Waksman, P.: The first eigenvalue of a scalene triangle. Proc. Am. Math. Soc. 100(1), 175–182 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bucur, D., Varchon, N.: Boundary variation for a Neumann problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(4), 807–821 (2000)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  6. Bueno, H., Ercole, G.: Solutions of the Cheeger problem via torsion functions. J. Math. Anal. Appl. 381(1), 263–279 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities, Fundamental Principles of Mathematical Sciences, vol. 285. Springer, Berlin (1988)

    Book  Google Scholar 

  8. Buttazzo, G., Carlier, G., Comte, M.: On the selection of maximal Cheeger sets. Differ. Integr. Equ. 20(9), 991–1004 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Carlier, G., Comte, M.: On a weighted total variation minimization problem. J. Funct. Anal. 250(1), 214–226 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caselles, V., Chambolle, A., Novaga, M.: Uniqueness of the Cheeger set of a convex body. Pac. J. Math. 232(1), 77–90 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969). Princeton University Press, Princeton (1970)

    Google Scholar 

  12. Crasta, G., Fragalà, I., Gazzola, F.: A sharp upper bound for the torsional rigidity of rods by means of web functions. Arch. Ration. Mech. Anal. 164(3), 189–211 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Evans, L.C.: The 1-Laplacian, the \(\infty \)-Laplacian and differential games. Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, vol. 446, pp. 245–254. American Mathematical Society, Providence (2007)

    Chapter  Google Scholar 

  14. Figalli, A., Maggi, F., Pratelli, A.: A note on Cheeger sets. Proc. Am. Math. Soc. 137(6), 2057–2062 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fragalà, I., Gazzola, F., Lamboley, J.: Some sharp bounds for the p-torsion of convex domains. Geometric Properties for Parabolic and Elliptic PDE’s, vol. 2, pp. 97–115. Springer INdAM Series, Berlin (2013)

    Chapter  Google Scholar 

  16. Freitas, P., Siudeja, B.: Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. ESAIM Control Optim. Calc. Var. 16(3), 648–676 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fridman, V., Kawohl, B.: Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44(4), 659–667 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Fusco, N., Maggi, F., Pratelli, A.: Stability estimates for certain Faber–Krahn, isocapacitary and Cheeger inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(1), 51–71 (2009)

    MATH  MathSciNet  Google Scholar 

  19. Henrot, A., Pierre, M.: Variation et optimisation de formes. Mathématiques & Applications (Berlin) [Mathematics & Applications]. Une analyse géométrique. [A geometric analysis], vol. 48. Springer, Berlin (2005)

    Google Scholar 

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

    Google Scholar 

  21. Juutinen, P., Lindqvist, P., Manfredi, J.J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kawohl, B., Lachand-Robert, T.: Characterization of Cheeger sets for convex subsets of the plane. Pac. J. Math. 225(1), 103–118 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kawohl, B., Schuricht, F.: Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9(4), 515–543 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kawohl, B., Novaga, M.: The \(p\)-Laplace eigenvalue problem as \(p\rightarrow 1\) and Cheeger sets in a Finsler metric. J. Convex Anal. 15(3), 623–634 (2008)

    MATH  MathSciNet  Google Scholar 

  25. Krejčiřík, D., Pratelli, A.: The Cheeger constant of curved strips. Pac. J. Math. 254(2), 309–333 (2011)

    Article  MATH  Google Scholar 

  26. Parini, E.: An introduction to the Cheeger problem. Surv. Math. Appl. 6, 9–21 (2011)

    MathSciNet  Google Scholar 

  27. Pólya, G., Szegő, G.: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol. 27. Princeton University Press, Princeton (1951)

    Google Scholar 

  28. Solynin, A.Y., Zalgaller, V.A.: An isoperimetric inequality for logarithmic capacity of polygons. Ann. Math. (2) 159(1), 277–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stredulinsky, E., Ziemer, W.P.: Area minimizing sets subject to a volume constraint in a convex set. J. Geom. Anal. 7(4), 653–677 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the ANR-12-BS01-0007 Optiform and GNAMPA (INDAM), and was finished during the first author’s visit at the Isaac Newton Institute for Mathematical Sciences, Cambridge, within the programme “Free Boundary Problems and Related Topics”.

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Correspondence to Dorin Bucur.

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Communicated by Jiaping Wang.

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Bucur, D., Fragalà, I. A Faber–Krahn Inequality for the Cheeger Constant of \(N\)-gons. J Geom Anal 26, 88–117 (2016). https://doi.org/10.1007/s12220-014-9539-5

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  • DOI: https://doi.org/10.1007/s12220-014-9539-5

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