On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique
نویسندگان
چکیده
We generalize to the p−Laplacian ∆p 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 ∆p 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 having 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.
منابع مشابه
Analysis of Stress Intensity Factors in Hollow Cylinders Reinforced by an Effective Coating Containing Multiple Cracks
In this paper, the solution of an isotropic hollow cylinder, with an isotropic coating, weakened by multiple radial cracks is studied. The hollow cylinder and its coating are under Saint-Venant torsional loading. The series solution is then derived for displacement and stress fields in the cross section of the cylinder containing a Volterra-type screw dislocation. The dislocation solution is em...
متن کاملSpectral Optimization Problems for Potentials and Measures
In the present paper we consider spectral optimization problems involving the Schrödinger operator −∆ + μ on R, the prototype being the minimization of the k the eigenvalue λk(μ). Here μ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential V which satisfies the integral constraint ∫ V −pdx ≤ m with 0 < p < 1. We p...
متن کاملTorsional Surface Wave Propagation in Anisotropic Layer Sandwiched Between Heterogeneous Half-Space
The present paper studies the possibility of propagation of torsional surface waves in an inhomogeneous anisotropic layer lying between two heterogeneous half-spaces (upper and lower half-space). Both the half-spaces are assumed to be under compressive initial stress. The study reveals that under the assumed conditions, a torsional surface wave propagates in the medium. The dispersion relation ...
متن کاملSurface Energy and Elastic Medium Effects on Torsional Vibrational Behavior of Embedded Nanorods
In this paper surface energy and elastic medium effects on torsional vibrational behavior of nanorods are studied. The surface elasticity theory is used to consider the surface energy effects and the elastic medium is modeled as torsional springs attached to the nanorod. At the next step, Hamilton’s principle is utilized to derive governing equations and boundary conditions. Then, with the aid ...
متن کاملDispersion of Torsional Surface Wave in a Pre-Stressed Heterogeneous Layer Sandwiched Between Anisotropic Porous Half-Spaces Under Gravity
The study of surface waves in a layered media has their viable application in geophysical prospecting. This paper presents an analytical study on the dispersion of torsional surface wave in a pre-stressed heterogeneous layer sandwiched between a pre-stressed anisotropic porous semi-infinite medium and gravitating anisotropic porous half-space. The non-homogeneity within the intermediate layer a...
متن کامل