Isoperimetric estimates for the first eigenvalue of the p -Laplace operator and the Cheeger constant

Bernhard Kawohl; V. Fridman

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 659-667
  • ISSN: 0010-2628

Abstract

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First we recall a Faber-Krahn type inequality and an estimate for λ p ( Ω ) in terms of the so-called Cheeger constant. Then we prove that the eigenvalue λ p ( Ω ) converges to the Cheeger constant h ( Ω ) as p → 1 . The associated eigenfunction u p converges to the characteristic function of the Cheeger set, i.e. a subset of Ω which minimizes the ratio | ∂ D | / | D | among all simply connected D ⊂ ⊂ Ω . As a byproduct we prove that for convex Ω the Cheeger set ω is also convex.

How to cite

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Kawohl, Bernhard, and Fridman, V.. "Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 659-667. <http://eudml.org/doc/249207>.

@article{Kawohl2003,
abstract = {First we recall a Faber-Krahn type inequality and an estimate for $\lambda _p(\Omega )$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue $\lambda _p(\Omega )$ converges to the Cheeger constant $h(\Omega )$ as $p\rightarrow 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega $ which minimizes the ratio $|\partial D|/|D|$ among all simply connected $D\subset \subset \Omega $. As a byproduct we prove that for convex $\Omega $ the Cheeger set $\omega $ is also convex.},
author = {Kawohl, Bernhard, Fridman, V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator; Faber-Krahn type inequality; eigenvalue for -Laplacian; Cheeger set; 1-Laplace operator},
language = {eng},
number = {4},
pages = {659-667},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant},
url = {http://eudml.org/doc/249207},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Kawohl, Bernhard
AU - Fridman, V.
TI - Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 659
EP - 667
AB - First we recall a Faber-Krahn type inequality and an estimate for $\lambda _p(\Omega )$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue $\lambda _p(\Omega )$ converges to the Cheeger constant $h(\Omega )$ as $p\rightarrow 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega $ which minimizes the ratio $|\partial D|/|D|$ among all simply connected $D\subset \subset \Omega $. As a byproduct we prove that for convex $\Omega $ the Cheeger set $\omega $ is also convex.
LA - eng
KW - isoperimetric estimates; eigenvalue; Cheeger constant; $p$-Laplace operator; $1$-Laplace operator; Faber-Krahn type inequality; eigenvalue for -Laplacian; Cheeger set; 1-Laplace operator
UR - http://eudml.org/doc/249207
ER -

References

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  1. Alvino A., Ferone V., Trombetti G., On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal. 29 (1998), 437-451. (1998) Zbl0908.35094MR1616519
  2. Bhattacharia T., A proof of the Faber-Krahn inequality for the first eigenvalue of the p -Laplacian, Ann. Mat. Pura Appl. Ser. 4 177 (1999), 225-240. (1999) MR1747632
  3. Belloni M., Kawohl B., A direct uniqueness proof for equations involving the p -Laplace operator, Manuscripta Math. 109 (2002), 229-231. (2002) Zbl1100.35032MR1935031
  4. Belloni M., Kawohl B., The pseudo- p -Laplace eigenvalue problem and viscosity solutions as p → ∞ , ESAIM COCV, to appear. Zbl1092.35074MR2084254
  5. Chavel I., Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives, Cambridge University Press, Cambridge, 2001. Zbl0988.51019MR1849187
  6. Cheeger J., A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis, A Symposium in Honor of Salomon Bochner, R.C. Gunning, Ed., Princeton Univ. Press, 1970, pp.195-199. Zbl0212.44903MR0402831
  7. Cicaclese M., Trombetti C., Asymptotic behaviour of solutions to p -Laplacian equation, preprint No. 13, Univ. Napoli, 2002. 
  8. Demengel F., Theorémès d’existence pour des équations avec l’opérateur 1 -Laplacien, première valeur propre pour - Δ 1 , C.R. Acad. Sci. Paris, Ser. I 334 (2002), 1071-1076. (2002) Zbl1142.35408MR1911649
  9. Fridman V., doctoral thesis, in preparation, . 
  10. Giusti E., Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. Zbl0825.49059MR0775682
  11. Gonzales E., Massari U., Tamanini I., On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), 25-37. (1983) MR0684753
  12. Huang Y.X., On the eigenvalues of the p -Laplacian with varying p , Proc. Amer. Math. Soc. 125 (1997), 3347-3354. (1997) Zbl0882.35087MR1403133
  13. Juutinen P., Lindqvist P., Manfredi J., The ∞ -eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), 89-105. (1999) Zbl0947.35104MR1716563
  14. Kawohl B., Rearrangements and Convexity of Level Sets in PDE, Springer Lecture Notes in Math. 1150, 1985. Zbl0593.35002MR0810619
  15. Kawohl B., On a family of torsional creep problems, J. reine angew. Math. 410 (1990), 1-22. (1990) Zbl0701.35015MR1068797
  16. Kawohl B., Stará J., Wittum G., Analysis and numerical studies of a problem of shape design, Arch. Ration. Mech. Anal. 114 (1991), 349-363. (1991) MR1100800
  17. Kawohl B., Kutev N., Global behavior of solutions to a parabolic mean curvature equation, Differ. Integral Equations 8 (1995), 1923-1946. (1995) MR1348958
  18. Kawohl B., Some nonconvex shape optimization problems, in: Optimal Shape Design, B. Kawohl et al., Eds., Springer Lecture Notes in Math. 1740 (2000), 7-46. Zbl0982.49024MR1804684
  19. Lindqvist P., On non-linear Rayleigh quotients, Potential Anal. 2 (1993), 199-218. (1993) 
  20. Lindqvist P., A note on the nonlinear Rayleigh quotient, in: Analysis, Algebra and Computers in Mathematical Research (Lulea 1992), M. Gyllenberg & L.E. Persson, Eds., Marcel Dekker Lecture Notes in Pure and Appl. Math. 156, 1994, pp. 223-231. Zbl0805.35085MR1280948
  21. Lindqvist P., On a nonlinear eigenvalue problem, Padova, 2000, pp. 79-110. Zbl0838.35094
  22. Lefton L., Wei D., Numerical approximation of the first eigenpair of the p -Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optim. 18 (1997), 389-399. (1997) Zbl0884.65103MR1448898
  23. Matei A.M., First eigenvalue for the p -Laplace operator, Nonlinear Anal. TMA 39 (2000), 1051-1061. (2000) Zbl0948.35090MR1735181
  24. Marcellini P., Miller K., Elliptic versus parabolic regularization for the equation of prescribed mean curvature, J. Differential Equations 137 (1997), 1-53. (1997) Zbl0890.35046MR1451535
  25. Payne L.E., Rayner M.E., An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z. Angew. Math. Phys. 23 (1972), 13-15. (1972) Zbl0241.73080MR0313649
  26. Payne L.E., Rayner M.E., Some isoperimetric norm bound for solutions of the Helmholtz equation, Z. Angew. Math. Phys. 24 (1973), 105-110. (1973) MR0324202
  27. Sakaguchi S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sci. Norm. Sup. Pisa (IV) 14 (1987), 404-421. (1987) Zbl0665.35025MR0951227
  28. Shirakawa K., Asymptotic convergence of p -Laplace equations with constraints as p tends to 1 , Math. Methods Appl. Sci. 25 (2002), 771-793. (2002) MR1906858
  29. Stredulinsky E., Ziemer W.P., Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Anal. 7 (1997), 653-677. (1997) Zbl0940.49025MR1669207

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