Abstract
We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.
Funding statement: Nikita Simonov was partially supported by the Spanish Ministry of Science and Innovation, through the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Ministry of Science and Innovation, Spain), by the project MTM2017-85757-P (Ministry of Science and Innovation, Spain), by the E.U. H2020 MSCA programme, grant agreement 777822, by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR), and by the DIM Math-Innov of the Region Île-de-France.
Acknowledgements
A part of this project was carried out in Univesidad Autónoma de Madrid, when Iwona Chlebicka was visiting Matteo Bonforte. Both authors are grateful to him for guidance, patience, and invaluable help. Additionally, Iwona Chlebicka would like to thank Michał Strzelecki for insightful discussions and Błażej Miasojedow for essential help with computations. The authors would like to express their gratitude for reviewers who provided deep comments that substantially helped the presentation of the paper.
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Articles in the same Issue
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow
Articles in the same Issue
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow