On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique

Lorenzo Brasco

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 315-338
  • ISSN: 1292-8119

Abstract

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We generalize to the p-Laplacian Δp a spectral inequality proved by M.-T. Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound on the first Dirichlet eigenvalue of Δp of a set in terms of its p-torsional rigidity. The result is valid in every space dimension, for every 1 < p < ∞ and for every open set with finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincaré-Sobolev constants. The method of proof is based on a generalization of the rearrangement technique introduced by Kohler−Jobin.

How to cite

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Brasco, Lorenzo. "On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 315-338. <http://eudml.org/doc/272938>.

@article{Brasco2014,
abstract = {We generalize to the p-Laplacian Δp a spectral inequality proved by M.-T. Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound on the first Dirichlet eigenvalue of Δp of a set in terms of its p-torsional rigidity. The result is valid in every space dimension, for every 1 &lt; p &lt; ∞ and for every open set with finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincaré-Sobolev constants. The method of proof is based on a generalization of the rearrangement technique introduced by Kohler−Jobin.},
author = {Brasco, Lorenzo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {torsional rigidity; nonlinear eigenvalue problems; spherical rearrangements; -Laplacian},
language = {eng},
number = {2},
pages = {315-338},
publisher = {EDP-Sciences},
title = {On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique},
url = {http://eudml.org/doc/272938},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Brasco, Lorenzo
TI - On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 2
SP - 315
EP - 338
AB - We generalize to the p-Laplacian Δp a spectral inequality proved by M.-T. Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound on the first Dirichlet eigenvalue of Δp of a set in terms of its p-torsional rigidity. The result is valid in every space dimension, for every 1 &lt; p &lt; ∞ and for every open set with finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincaré-Sobolev constants. The method of proof is based on a generalization of the rearrangement technique introduced by Kohler−Jobin.
LA - eng
KW - torsional rigidity; nonlinear eigenvalue problems; spherical rearrangements; -Laplacian
UR - http://eudml.org/doc/272938
ER -

References

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  1. [1] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lect. Series Math. Appl. Oxford University Press, Oxford (2004). Zbl1080.28001MR2039660
  2. [2] A. Alvino, V. Ferone, P.-L. Lions and G. Trombetti, Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire14 (1997) 275–293. Zbl0877.35040MR1441395
  3. [3] M. Belloni and B. Kawohl, The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 28–52. Zbl1092.35074MR2084254
  4. [4] L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form, preprint (2013), available at http://cvgmt.sns.it/paper/2161/ 
  5. [5] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, vol. 65 of Progress Nonlinear Differ. Eqs. Birkhäuser Verlag, Basel (2005). Zbl1117.49001MR2150214
  6. [6] T. Carroll and J. Ratzkin, Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl.379 (2011) 818–826. Zbl1216.35016MR2784361
  7. [7] E. DiBenedetto, C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal.7 (1983) 827–850. Zbl0539.35027MR709038
  8. [8] I. Ekeland, Convexity methods in Hamiltonian mechanics. Springer-Verlag (1990). Zbl0707.70003MR1051888
  9. [9] L. Esposito and C. Trombetti, Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal.56 (2004) 43–62. Zbl1038.26014MR2031435
  10. [10] A. Ferone and R. Volpicelli, Convex rearrangement: equality cases in the Pólya-Szegő inequality, Calc. Var. Partial Differ. Eqs.21 (2004) 259–272. Zbl1116.49022MR2094322
  11. [11] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math.81 (2010) 167–211. Zbl1196.49033MR2672283
  12. [12] M. Flucher, Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv.67 (1992) 471–497. Zbl0763.58008MR1171306
  13. [13] I. Fragalà, F. Gazzola and J. Lamboley, Sharp bounds for the p-torsion of convex planar domains, in Geometric Properties for Parabolic and Elliptic PDE’s, vol. 2 of Springer INdAM Series (2013) 97–115. Zbl1278.49050MR3050229
  14. [14] G. Franzina, P. D. Lamberti, Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Eqs. (2010) 10. Zbl1188.35125MR2602859
  15. [15] N. Fusco, F. Maggi and A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sci. Norm. Super. Pisa Cl. Sci.8 (2009) 51–71. Zbl1176.49047MR2512200
  16. [16] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). Zbl1109.35081MR2251558
  17. [17] S. Kesavan, Symmetrization and applications, in vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). Zbl1110.35002MR2238193
  18. [18] M.-T. Kohler-Jobin, Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal.13 (1982), 153–161. Zbl0484.35006MR641547
  19. [19] M.-T. Kohler-Jobin, Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique. I. Une démonstration de la conjecture isopérimétrique P λ 2 ≥ π j 0 4 / 2 P λ2≥π j04/2 de Pólya et Szegő, Z. Angew. Math. Phys. 29 (1978) 757–766. Zbl0427.73056MR511908
  20. [20] M.-T. Kohler-Jobin, Démonstration de l’inégalité isopérimétrique P λ 2 ≥ π j 0 4 / 2 P λ2≥π j04/2, conjecturée par Pólya et Szegő. C.R. Acad. Sci. Paris Sér. A-B 281 (1975) A119–A121. Zbl0303.52003MR385691
  21. [21] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 1203–1219. Zbl0675.35042MR969499
  22. [22] K.-C. Lin, Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc.348 (1996) 2663–2671. Zbl0861.49001MR1333394
  23. [23] J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 1077–1092. Zbl0213.13001
  24. [24] G. Pólya, G. Szegő, Isoperimetric inequalities in mathematical physics, in vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, N. J. (1951). Zbl0044.38301
  25. [25] R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge University Press (1993). Zbl1287.52001MR1216521
  26. [26] G. Talenti, Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa3 (1976) 697–718. Zbl0341.35031MR601601
  27. [27] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications. J. Math. Mech.17 (1967), 473–483. Zbl0163.36402MR216286

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