Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities
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The first eigenvalue of the p-Laplacian on an open set of given mea- sure attains its minimum value if and only if the set is a ball as mentioned in this paper.Abstract:
The first eigenvalue of the p-Laplacian on an open set of given mea- sure attains its minimum value if and only if the set is a ball. We provide a quantitative version of this statement by an argument that can be easily adapted to treat also certain isocapacitary and Cheeger inequalities.read more
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A Selection Principle for the Sharp Quantitative Isoperimetric Inequality
TL;DR: In this paper, a variational method for the study of isoperimetric inequalities with quantitative terms is introduced. But the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter.
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Eigenvalues for double phase variational integrals
TL;DR: In this article, a sequence of nonlinear eigenvalues is introduced by a minimax procedure, and the growth rate of this sequence is investigated with respect to the variations of the phases.
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The sharp Sobolev inequality in quantitative form
TL;DR: A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals in this paper, where the distance is defined as
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A Selection Principle for the Sharp Quantitative Isoperimetric Inequality
TL;DR: In this article, a variational method for the study of stability in the isoperimetric inequality is introduced, which relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter.
Journal ArticleDOI
Faber-Krahn inequalities in sharp quantitative form
TL;DR: The Faber-Krahn inequality as mentioned in this paper states that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume.
References
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George Pólya,Gábor Szegő +1 more
TL;DR: Isoperimetric Inequalities in Mathematical Physics (AM-27) as mentioned in this paper is an excellent survey of the literature in this area. But it is not a complete collection.
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The isoperimetric inequality
TL;DR: For a survey of generalizations of the isoperimetric inequality, see as mentioned in this paper, where the main focus is on geometric versions and generalisations of the inequality, with emphasis on recent contributions.