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.
Classification: 35P30, 47A75, 49Q10
Keywords: torsional rigidity, nonlinear eigenvalue problems, spherical rearrangements
@article{COCV_2014__20_2_315_0,
author = {Brasco, Lorenzo},
title = {On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {2},
year = {2014},
pages = {315-338},
doi = {10.1051/cocv/2013065},
zbl = {1290.35160},
mrnumber = {3264206},
language = {en},
url = {http://www.numdam.org/item/COCV_2014__20_2_315_0}
}
Brasco, Lorenzo. On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 315-338. doi : 10.1051/cocv/2013065. http://www.numdam.org/item/COCV_2014__20_2_315_0/
[1] and , Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lect. Series Math. Appl. Oxford University Press, Oxford (2004). | MR 2039660 | Zbl 1080.28001
[2] , , and , Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire 14 (1997) 275-293. | Numdam | MR 1441395 | Zbl 0877.35040
[3] and , The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 28-52. | Numdam | MR 2084254 | Zbl 1092.35074
[4] , and , Faber-Krahn inequalities in sharp quantitative form, preprint (2013), available at http://cvgmt.sns.it/paper/2161/
[5] and , Variational Methods in Shape Optimization Problems, vol. 65 of Progress Nonlinear Differ. Eqs. Birkhäuser Verlag, Basel (2005). | MR 2150214 | Zbl 1117.49001
[6] and , Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379 (2011) 818-826. | MR 2784361 | Zbl 1216.35016
[7] , C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827-850. | MR 709038 | Zbl 0539.35027
[8] , Convexity methods in Hamiltonian mechanics. Springer-Verlag (1990). | MR 1051888 | Zbl 0707.70003
[9] and , Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal. 56 (2004) 43-62. | MR 2031435 | Zbl 1038.26014
[10] and , Convex rearrangement: equality cases in the Pólya-Szegő inequality, Calc. Var. Partial Differ. Eqs. 21 (2004) 259-272. | MR 2094322 | Zbl 1116.49022
[11] , and , A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 81 (2010) 167-211. | MR 2672283 | Zbl 1196.49033
[12] , Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv. 67 (1992) 471-497. | MR 1171306 | Zbl 0763.58008
[13] , and , 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. | MR 3050229 | Zbl 1278.49050
[14] , , Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Eqs. (2010) 10. | MR 2602859 | Zbl 1188.35125
[15] , and , Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sci. Norm. Super. Pisa Cl. Sci. 8 (2009) 51-71. | Numdam | MR 2512200 | Zbl 1176.49047
[16] , Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR 2251558 | Zbl 1109.35081
[17] , Symmetrization and applications, in vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). | MR 2238193 | Zbl 1110.35002
[18] , Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), 153-161. | MR 641547 | Zbl 0484.35006
[19] , Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique | MR 511908 | Zbl 0427.73056
[20] , Démonstration de l'inégalité isopérimétrique | MR 385691 | Zbl 0303.52003
[21] , Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 1203-1219. | MR 969499 | Zbl 0675.35042
[22] , Extremal functions for Moser's inequality. Trans. Amer. Math. Soc. 348 (1996) 2663-2671. | MR 1333394 | Zbl 0861.49001
[23] , A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 1077-1092. | Zbl 0213.13001
[24] , , Isoperimetric inequalities in mathematical physics, in vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, N. J. (1951). | Zbl 0044.38301
[25] , Convex bodies: the Brunn-Minkowski theory. Cambridge University Press (1993). | MR 1216521 | Zbl 1287.52001
[26] , Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3 (1976) 697-718. | Numdam | MR 601601 | Zbl 0341.35031
[27] , On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. | MR 216286 | Zbl 0163.36402
