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

Published online by Cambridge University Press:  06 February 2014

Lorenzo Brasco
Lorenzo Brasco*
Affiliation:
Laboratoire d’Analyse, Topologie, Probabilités, Aix-Marseille Université, 39 Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France. lorenzo.brasco@univ-amu.fr
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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 < 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.

Keywords

Torsional rigiditynonlinear eigenvalue problemsspherical rearrangements
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 315 - 338
Copyright
© EDP Sciences, SMAI, 2014

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References

L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lect. Series Math. Appl. Oxford University Press, Oxford (2004).
Alvino, A., Ferone, V., Lions, P.-L. and Trombetti, G., Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire 14 (1997) 275–293. Google Scholar
Belloni, M. and Kawohl, B., The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 28–52. Google Scholar
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/
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, vol. 65 of Progress Nonlinear Differ. Eqs. Birkhäuser Verlag, Basel (2005).
Carroll, T. and Ratzkin, J., Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379 (2011) 818–826. Google Scholar
DiBenedetto, E., C 1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827–850. Google Scholar
I. Ekeland, Convexity methods in Hamiltonian mechanics. Springer-Verlag (1990).
Esposito, L. and Trombetti, C., Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal. 56 (2004) 43–62. Google Scholar
Ferone, A. and Volpicelli, R., Convex rearrangement: equality cases in the Pólya-Szegő inequality, Calc. Var. Partial Differ. Eqs. 21 (2004) 259–272. Google Scholar
Figalli, A., Maggi, F. and Pratelli, A., A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 81 (2010) 167–211. Google Scholar
Flucher, M., Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv. 67 (1992) 471–497. Google Scholar
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.
G. Franzina, P. D. Lamberti, Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Eqs. (2010) 10.
Fusco, N., Maggi, F. and Pratelli, A., Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sci. Norm. Super. Pisa Cl. Sci. 8 (2009) 51–71. Google Scholar
A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006).
S. Kesavan, Symmetrization and applications, in vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006).
Kohler-Jobin, M.-T., Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), 153–161. Google Scholar
Kohler-Jobin, M.-T., Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique. I. Une démonstration de la conjecture isopérimétrique \hbox{$P\,\lambda^2\ge \pi \, j^4_0/2$}P λ2≥π j04/2 de Pólya et Szegő, Z. Angew. Math. Phys. 29 (1978) 757–766. Google Scholar
Kohler-Jobin, M.-T., Démonstration de l’inégalité isopérimétrique \hbox{$P\, \lambda^2\ge \pi\, j_0^4/2$}P λ2≥π j04/2, conjecturée par Pólya et Szegő. C.R. Acad. Sci. Paris Sér. A-B 281 (1975) A119–A121. Google Scholar
Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 1203–1219. Google Scholar
Lin, K.-C., Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 348 (1996) 2663–2671. Google Scholar
Moser, J., A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 1077–1092. Google Scholar
G. Pólya, G. Szegő, Isoperimetric inequalities in mathematical physics, in vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, N. J. (1951).
R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge University Press (1993).
Talenti, G., Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3 (1976) 697–718. Google Scholar
Trudinger, N.S., On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–483. Google Scholar