Sharp quantitative stability for isoperimetric inequalities with homogeneous weights
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- by E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton and J. Serra PDF
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Abstract:
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights.
Inspired by the proof of such isoperimetric inequalities through the ABP method (see X. Cabré, X. Ros-Oton, and J. Serra [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2971–2998]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov’s proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [Figalli, Maggi, and Pratelli [Invent. Math. 182 (2010), pp. 167–211] and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations.
As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
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Additional Information
- E. Cinti
- Affiliation: Dipartimento di Matematica, Alma Mater Studorium Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
- MR Author ID: 899645
- Email: eleonora.cinti5@unibo.it
- F. Glaudo
- Affiliation: ETH Zürich, Department of Mathematics., Rämistrasse 101, 8092 Zürich, Switzerland
- MR Author ID: 1300022
- Email: federico.glaudo@math.ethz.ch
- A. Pratelli
- Affiliation: Universitá di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56127 Pisa, Italy
- MR Author ID: 696068
- ORCID: 0000-0002-3670-6625
- Email: aldo.pratelli@dm.unipi.it
- X. Ros-Oton
- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland; ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain; and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 920237
- Email: xavier.ros-oton@math.uzh.ch
- J. Serra
- Affiliation: ETH Zürich, Department of Mathematics., Rämistrasse 101, 8092 Zürich, Switzerland
- MR Author ID: 983074
- Email: joaquim.serra@math.ethz.ch
- Received by editor(s): June 26, 2020
- Received by editor(s) in revised form: April 13, 2021
- Published electronically: January 12, 2022
- Additional Notes: The second and fifth authors received funding from the European Research Council under the Grant Agreement No. 721675 “Regularity and Stability in Partial Differential Equations (RSPDE)”. The fourth author received funding from the European Research Council under the Grant Agreement No. 801867 “Regularity and singularities in elliptic PDE (EllipticPDE)”. The first, fourth, and fifth authors were supported by grant MTM2017-84214-C2-1-P. The first author was partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1509-1550
- MSC (2020): Primary 49Q10; Secondary 49Q20, 28A75, 49K40
- DOI: https://doi.org/10.1090/tran/8525
- MathSciNet review: 4378069