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 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},
pages = {315--338},
year = {2014},
publisher = {EDP Sciences},
volume = {20},
number = {2},
doi = {10.1051/cocv/2013065},
mrnumber = {3264206},
zbl = {1290.35160},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2013065/}
}
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 SP - 315 EP - 338 VL - 20 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2013065/ DO - 10.1051/cocv/2013065 LA - en ID - COCV_2014__20_2_315_0 ER -
%0 Journal Article %A Brasco, Lorenzo %T On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 315-338 %V 20 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2013065/ %R 10.1051/cocv/2013065 %G en %F 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, Tome 20 (2014) no. 2, pp. 315-338. doi: 10.1051/cocv/2013065
[1] and , Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lect. Series Math. Appl. Oxford University Press, Oxford (2004). | Zbl | MR
[2] , , and , Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire 14 (1997) 275-293. | Zbl | MR | Numdam
[3] and , The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 28-52. | Zbl | MR | Numdam
[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). | Zbl | MR
[6] and , Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379 (2011) 818-826. | Zbl | MR
[7] , C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827-850. | Zbl | MR
[8] , Convexity methods in Hamiltonian mechanics. Springer-Verlag (1990). | Zbl | MR
[9] and , Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal. 56 (2004) 43-62. | Zbl | MR
[10] and , Convex rearrangement: equality cases in the Pólya-Szegő inequality, Calc. Var. Partial Differ. Eqs. 21 (2004) 259-272. | Zbl | MR
[11] , and , A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 81 (2010) 167-211. | Zbl | MR
[12] , Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv. 67 (1992) 471-497. | Zbl | MR
[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. | Zbl | MR
[14] , , Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Eqs. (2010) 10. | Zbl | MR
[15] , and , Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sci. Norm. Super. Pisa Cl. Sci. 8 (2009) 51-71. | Zbl | MR | Numdam
[16] , Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | Zbl | MR
[17] , Symmetrization and applications, in vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). | Zbl | MR
[18] , Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), 153-161. | Zbl | MR
[19] , Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique | Zbl | MR
[20] , Démonstration de l'inégalité isopérimétrique | Zbl | MR
[21] , Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 1203-1219. | Zbl | MR
[22] , Extremal functions for Moser's inequality. Trans. Amer. Math. Soc. 348 (1996) 2663-2671. | Zbl | MR
[23] , A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 1077-1092. | Zbl
[24] , , Isoperimetric inequalities in mathematical physics, in vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, N. J. (1951). | Zbl
[25] , Convex bodies: the Brunn-Minkowski theory. Cambridge University Press (1993). | Zbl | MR
[26] , Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3 (1976) 697-718. | Zbl | MR | Numdam
[27] , On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. | Zbl | MR
Cité par Sources :
