The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.
Keywords: Non-local isoperimetric problem, optimal transport, existence of minimizers
@article{COCV_2022__28_1_A37_0,
author = {Candau-Tilh, Jules and Goldman, Michael},
title = {Existence and stability results for an isoperimetric problem with a non-local interaction of {Wasserstein} type},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022040},
mrnumber = {4438713},
zbl = {1526.49026},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022040/}
}
TY - JOUR AU - Candau-Tilh, Jules AU - Goldman, Michael TI - Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022040/ DO - 10.1051/cocv/2022040 LA - en ID - COCV_2022__28_1_A37_0 ER -
%0 Journal Article %A Candau-Tilh, Jules %A Goldman, Michael %T Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022040/ %R 10.1051/cocv/2022040 %G en %F COCV_2022__28_1_A37_0
Candau-Tilh, Jules; Goldman, Michael. Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 37. doi: 10.1051/cocv/2022040
[1] , and , Minimality via second variation for a nonlocal isoperimetric problem. Commun. Math. Phys. 322 (2013) 515–557. | MR | Zbl | DOI
[2] , and , On the Wasserstein distance between mutually singular measures. Adv. Calc. Var. 13 (2020) 141–154. | MR | Zbl | DOI
[3] , and , Minimality of balls in the small volume regime for a general Gamow type functional (2020). | Zbl
[4] and , Indeterminacy estimates and the size of nodal sets in singular spaces. Adv. Math. 389 (2021) Paper No. 107919, 38. | MR | Zbl | DOI
[5] , and , An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications. Notices Amer. Math. Soc. 64 (2017) 1275–1283. | MR | Zbl | DOI
[6] and , A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206 (2012) 617–643. | MR | Zbl | DOI
[7] , , and , BV estimates in optimal transportation and applications. Arch. Rati. Mech. Anal. 219 (2015) 829–860. | MR | Zbl | DOI
[8] , , , and , Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336 (2015) 441–507. | MR | Zbl | DOI
[9] 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
[10] and , Existence and nonexistence in the liquid drop model (2021). | MR | Zbl
[11] , Stability in the isoperimetric problem for convex or nearly spherical domains in . Trans. Am. Math. Soc. 314 (1989). | MR | Zbl
[12] and , Volume-constrained minimizers for the prescribed curvature problem in periodic media. Calc. Var. Partial Differ. Equ. 44 (2012) 297–318. | MR | Zbl | DOI
[13] , and , Existence and stability for a non-local isoperimetric model of charged liquid drops. Arch. Ration. Mech. Anal. 217 (2015) 1–36. | MR | Zbl | DOI
[14] 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
[15] , and , Low density phases in a uniformly charged liquid. Commun. Math. Phys. 345 (2016) 141–183. | MR | Zbl | DOI
[16] , Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press (2012). | MR | Zbl | DOI
[17] and , Minimality of the ball for a model of charged liquid droplets (2019). | Zbl
[18] , and , Least Wasserstein distance between disjoint shapes with perimeter regularization. Preprints (2021). | arXiv | MR | Zbl
[19] and , Minimisers of a general Riesz-type problem. Nonlinear Anal. 209 (2021) Paper No. 112346, 27. | MR | Zbl | DOI
[20] , Large mass minimizers for isoperimetric problems with integrable nonlocal potentials. Nonlinear Anal. 211 (2021) Paper No. 112395, 48. | MR | Zbl | DOI
[21] and , Partial localization, lipid bilayers, and the elastica functional. Arch. Ration. Mech. Anal. 193 (2009) 475–537. | MR | Zbl | DOI
[22] , Ensembles quasi-minimaux avec contrainte de volume et rectifiabilité uniforme. Mém. Soc. Math. Fr. (N.S.) (2000) vi+104. | MR | Zbl | Numdam
[23] , Optimal transport for applied mathematicians. Vol. 87 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, Cham (2015). | MR | Zbl | DOI
[24] , Topics in optimal transportation. Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003). | MR | Zbl
[25] and , The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains. Adv. Calc. Variat. (2021) 000010151520200083. | MR | Zbl
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