We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is 1∕2 and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.
Keywords: Riesz rearrangement inequality, fractional perimeter, Riesz potential, quantitative isoperimetric inequality
@article{COCV_2022__28_1_A4_0,
author = {Ascione, Giacomo},
title = {A spherical rearrangement proof of the stability of a {Riesz-type} inequality and an application to an isoperimetric type problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2021106},
mrnumber = {4362197},
zbl = {1481.49041},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021106/}
}
TY - JOUR AU - Ascione, Giacomo TI - A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021106/ DO - 10.1051/cocv/2021106 LA - en ID - COCV_2022__28_1_A4_0 ER -
%0 Journal Article %A Ascione, Giacomo %T A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021106/ %R 10.1051/cocv/2021106 %G en %F COCV_2022__28_1_A4_0
Ascione, Giacomo. A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 4. doi: 10.1051/cocv/2021106
[1] , Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces. Springer (2003) 1–52. | MR | Zbl
[2] , Convergence of Probability Measures. John Wiley & Sons (2013). | MR | Zbl
[3] and , Local and global minimality results for a nonlocal isoperimetric problem on . SIAM J. Math. Anal. 46 (2014) 2310–2349. | MR | Zbl | DOI
[4] , An Introduction to the Theory of Infinite Series. Vol. 335. Merchant Books (1908). | JFM
[5] , Cases of equality in the Riesz rearrangement inequality. Ann. Math. (1996) 499–527. | MR | Zbl | DOI
[6] and , Geometric stability of the Coulomb energy. Calc. Variat. Partial Differ. Equ. 54 (2015) 3241–3250. | MR | Zbl | DOI
[7] , and , Nonlocal shape optimization via interactions of attractive and repulsive potentials. Indiana Univ. Math. J. 67 (2018) 375–395. | MR | Zbl | DOI
[8] , and , On minimizers of interaction functionals with competing attractive and repulsive potentials, vol. 32 of Annales de l’Institut Henri Poincare (C) Non Linear Analysis. Elsevier (2015) 1283–1305. | MR | Zbl | Numdam | DOI
[9] , and , An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications. Notic. AMS 64 (2017) 1275–1283. | MR | Zbl
[10] and , Small volume-fraction limit of the diblock copolymer problem: II. Diffuse-interface functional. SIAM J. Math. Anal. 43 (2011) 739–763. | MR | Zbl | DOI
[11] , A sharpened Riesz-Sobolev inequality. Preprint (2017). | arXiv
[12] and , A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ratl. Mech. Anal. 206 (2012) 617–643. | MR | Zbl | DOI
[13] and , Regularity theory for local and nonlocal minimal surfaces: an overview, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Springer (2017) 117–158. | MR | Zbl | DOI
[14] and , Probabilities and potential. Bull. Am. Math. Soc. 2 (1980) 510–514. | DOI
[15] , The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions. Boletim da Sociedade Paranaense de Matemática 37 (2017) 143–157. | MR | Zbl | DOI
[16] and , Measure Theory and Fine Properties of Functions. CRC Press (2015). | MR | Zbl | DOI
[17] , Weyl and Marchaud derivatives: a forgotten history. Mathematics 6 (2018) 6. | Zbl | DOI
[18] , , , and , Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336 (2015) 441–507. | MR | Zbl | DOI
[19] , and , Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21 (2008) 925–950. | MR | Zbl | DOI
[20] , and , Nonexistence of large nuclei in the liquid drop model. Lett. Math. Phys. 106 (2016) 1033–1036. | MR | Zbl | DOI
[21] and , A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47 (2015) 4436–4450. | MR | Zbl | DOI
[22] and , A note on a theorem of M. Christ. Preprint (2019). | arXiv
[23] and , Proof of spherical flocking based on quantitative rearrangement inequalities. Preprint (2019). | arXiv | MR | Zbl
[24] and , Existence and nonexistence in the liquid drop model. Calc. Variat. Partial Differ. Equ. 60 (2021). | MR | Zbl
[25] , and , The ionization conjecture in Thomas–Fermi–Dirac–von Weizsäcker theory. Commun. Pure Appl. Math. 71 (2018) 577–614. | MR | Zbl | DOI
[26] , Stability in the isoperimetric problem for convex or nearly spherical domains in . Trans. Am. Math. Soc. 314 (1989) 619–638. | MR | Zbl
[27] , The quantitative isoperimetric inequality and related topics. Bull. Math. Sci. 5 (2015) 517–607. | MR | Zbl | DOI
[28] and , Sharp stability for the Riesz potential. ESAIM: COCV 26 (2020) 113. | MR | Zbl | Numdam
[29] , Mass defect curve and nuclear constitution. Proc. Royal Soc. London A 126 (1930) 632–644. | JFM
[30] and , Table of Integrals, Series, and Products. Academic Press (2014).
[31] , Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J. (2014) 77–89. | MR | Zbl | DOI
[32] and , On an isoperimetric problem with a competing nonlocal term I: the planar case. Commun. Pure Appl. Math. 66 (2013) 1129–1162. | MR | Zbl | DOI
[33] and , On an isoperimetric problem with a competing nonlocal term II: the general case. Commun. Pure Appl. Math. 67 (2014) 1974–1994. | MR | Zbl | DOI
[34] An isoperimetric problem with a Coulombic repulsion and attractive term. ESAIM: COCV 25 (2019) 14. | MR | Zbl | Numdam
[35] , The concentration-compactness principle in the Calculus of Variations. The locally compact case. Part 1. Ann. l’Institut Henri Poincaré (C) Non Linear Analysis 1 (1984) 109–145. | MR | Zbl | Numdam | DOI
[36] and , Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model. Commun. Pure Appl. Math. 67 (2014) 1605–1617. | MR | Zbl | DOI
[37] and , An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus. Preprint (2015). | arXiv | Numdam | MR | Zbl
[38] , Maximum and minimum sets for some geometric mean values. J. Theor. Probab. 3 (1990) 169–179. | MR | Zbl | DOI
[39] , , and , Function Spaces 1. Walter de Gruyter (2012). | DOI
[40] , Sur une inegalite integarale. J. London Math. Soc. 1 (1930) 162–168. | MR | JFM | DOI
[41] , The inversion of fractional integrals on a sphere. Israel J. Math. 79 (1992) 47–81. | MR | Zbl | DOI
[42] , Vol. 160 of Introduction to Radon transforms. Cambridge University Press (2015). | MR | Zbl
[43] , Hypersingular Integrals and Their Applications. CRC Press (2001). | MR | DOI
[44] , Generalized Hypergeometric Functions. Cambridge Univ. Press (1966). | MR | Zbl
[45] , et al., The asymptotic expansion of a ratio of gamma functions. Pacific J. Math. 1 (1951) 133–142. | MR | Zbl | DOI
[46] , Vol. 338 of Optimal Transport: Old and New. Springer Science & Business Media (2008). | MR | Zbl
Cité par Sources :
The author is supported by MIUR-PRIN 2017, project Stochastic Models for Complex Systems, no. 2017JFFHSH and by Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA-INdAM).
