On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω , we consider the problem of providing sharp lower bounds for λ 2 ( Ω ), i.e. its second Dirichlet eigenvalue of the p -Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality , asserting that the disjoint unions of two equal balls minimize λ...

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Bibliographic Details
Published in:Manuscripta mathematica 2013-07, Vol.141 (3-4), p.537-557
Main Authors: Brasco, Lorenzo, Franzina, Giovanni
Format: Article
Language:English
Subjects:
Algebraic Geometry
Calculus of Variations and Optimal Control
Optimization
Geometry
Lie Groups
Mathematics
Mathematics and Statistics
Number Theory
Topological Groups
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Summary:Given an open set Ω , we consider the problem of providing sharp lower bounds for λ 2 ( Ω ), i.e. its second Dirichlet eigenvalue of the p -Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality , asserting that the disjoint unions of two equal balls minimize λ 2 among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.
ISSN:0025-2611
1432-1785
DOI:10.1007/s00229-012-0582-x