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
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}, zbl = {1290.35160}, mrnumber = {3264206}, language = {en}, url = {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, Tome 20 (2014) no. 2, pp. 315-338. doi : 10.1051/cocv/2013065. http://www.numdam.org/item/COCV_2014__20_2_315_0/
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