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.

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.

DOI : 10.1051/cocv/2013065
Classification : 35P30, 47A75, 49Q10
Mots clés : 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},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {2},
     year = {2014},
     doi = {10.1051/cocv/2013065},
     mrnumber = {3264206},
     zbl = {1290.35160},
     language = {en},
     url = {http://archive.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  - http://archive.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 http://archive.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. http://archive.numdam.org/articles/10.1051/cocv/2013065/

[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). | MR | Zbl

[2] A. Alvino, V. Ferone, P.-L. Lions and G. Trombetti, Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire 14 (1997) 275-293. | Numdam | MR | Zbl

[3] M. Belloni and B. Kawohl, The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 28-52. | Numdam | MR | Zbl

[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] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, vol. 65 of Progress Nonlinear Differ. Eqs. Birkhäuser Verlag, Basel (2005). | MR | Zbl

[6] T. Carroll and J. Ratzkin, Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379 (2011) 818-826. | MR | Zbl

[7] E. Dibenedetto, C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827-850. | MR | Zbl

[8] I. Ekeland, Convexity methods in Hamiltonian mechanics. Springer-Verlag (1990). | MR | Zbl

[9] L. Esposito and C. Trombetti, Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal. 56 (2004) 43-62. | MR | Zbl

[10] A. Ferone and R. Volpicelli, Convex rearrangement: equality cases in the Pólya-Szegő inequality, Calc. Var. Partial Differ. Eqs. 21 (2004) 259-272. | MR | Zbl

[11] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 81 (2010) 167-211. | MR | Zbl

[12] M. Flucher, Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv. 67 (1992) 471-497. | MR | Zbl

[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. | MR | Zbl

[14] G. Franzina, P. D. Lamberti, Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differ. Eqs. (2010) 10. | MR | Zbl

[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. | Numdam | MR | Zbl

[16] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR | Zbl

[17] S. Kesavan, Symmetrization and applications, in vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). | MR | Zbl

[18] M.-T. Kohler-Jobin, Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), 153-161. | MR | Zbl

[19] M.-T. Kohler-Jobin, Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique | MR | Zbl

[20] M.-T. Kohler-Jobin, Démonstration de l'inégalité isopérimétrique | MR | Zbl

[21] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 1203-1219. | MR | Zbl

[22] K.-C. Lin, Extremal functions for Moser's inequality. Trans. Amer. Math. Soc. 348 (1996) 2663-2671. | MR | Zbl

[23] J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 1077-1092. | Zbl

[24] G. Pólya, G. Szegő, Isoperimetric inequalities in mathematical physics, in vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, N. J. (1951). | Zbl

[25] R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge University Press (1993). | MR | Zbl

[26] G. Talenti, Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3 (1976) 697-718. | Numdam | MR | Zbl

[27] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. | MR | Zbl

Cité par Sources :