Abstract
In this article, we consider the (double) minimization problem
where
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Communicated by: Frank Duzaar
References
[1] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, 1–199. 10.1090/memo/0165Search in Google Scholar
[2] L. Ambrosio, Lecture notes on optimal transport problems, Mathematical Aspects of Evolving Interfaces, Lecture Notes in Math. 1812, Springer, Berlin (2003), 1–52. 10.1007/978-3-540-39189-0_1Search in Google Scholar
[3] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. 10.1002/cpa.3160440402Search in Google Scholar
[4] G. Buttazzo, G. Carlier and M. Laborde, On the Wasserstein distance between mutually singular measures, Adv. Calc. Var. 13 (2020), no. 2, 141–154. 10.1515/acv-2017-0036Search in Google Scholar
[5] G. De Philippis, A. R. Mészáros, F. Santambrogio and B. Velichkov, BV estimates in optimal transportation and applications, Arch. Ration. Mech. Anal. 219 (2016), no. 2, 829–860. 10.1007/s00205-015-0909-3Search in Google Scholar
[6] S. Di Marino and F. Santambrogio, JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: Lp and Sobolev bounds, preprint (2019), https://arxiv.org/abs/1911.10999. 10.4171/aihpc/36Search in Google Scholar
[7] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar
[8] A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys. 336 (2015), no. 1, 441–507. 10.1007/s00220-014-2244-1Search in Google Scholar
[9] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010), no. 1, 167–211. 10.1007/s00222-010-0261-zSearch in Google Scholar
[10] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161. 10.1007/BF02392620Search in Google Scholar
[11] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29 (1998), no. 1, 1–17. 10.1137/S0036141096303359Search in Google Scholar
[12] H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term I: The planar case, Comm. Pure Appl. Math. 66 (2013), no. 7, 1129–1162. 10.1002/cpa.21451Search in Google Scholar
[13] H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term II: The general case, Comm. Pure Appl. Math. 67 (2014), no. 12, 1974–1994. 10.1002/cpa.21479Search in Google Scholar
[14] L. Lussardi, M. A. Peletier and M. Röger, Variational analysis of a mesoscale model for bilayer membranes, J. Fixed Point Theory Appl. 15 (2014), no. 1, 217–240. 10.1007/s11784-014-0180-5Search in Google Scholar
[15] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar
[16] E. Milakis, On the regularity of optimal sets in mass transfer problems, Comm. Partial Differential Equations 31 (2006), no. 4–6, 817–826. 10.1080/03605300500481244Search in Google Scholar
[17] F. Morgan, What is a surface?, Amer. Math. Monthly 103 (1996), no. 5, 369–376. 10.1080/00029890.1996.12004755Search in Google Scholar
[18] M. A. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional, Arch. Ration. Mech. Anal. 193 (2009), no. 3, 475–537. 10.1007/s00205-008-0150-4Search in Google Scholar
[19] F. Santambrogio, Optimal Transport for Applied Mathematicians, Progr. Nonlinear Differential Equations Appl. 87, Birkhäuser/Springer, Cham, 2015. 10.1007/978-3-319-20828-2Search in Google Scholar
[20] F. Santambrogio, {Euclidean, metric, and Wasserstein} gradient flows: An overview, Bull. Math. Sci. 7 (2017), no. 1, 87–154. 10.1007/s13373-017-0101-1Search in Google Scholar
[21] F. Santambrogio, Crowd motion and evolution PDEs under density constraints, SMAI 2017—8e Biennale Française des Mathématiques Appliquées et Industrielles, ESAIM Proc. Surveys 64, EDP Sciences, Les Ulis (2018), 137–157. 10.1051/proc/201864137Search in Google Scholar
[22] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, American Mathematical Society, Providence, 2003. 10.1090/gsm/058Search in Google Scholar
[23] C. Villani, Optimal Transport, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009. 10.1007/978-3-540-71050-9Search in Google Scholar
[24] Q. Xia, Regularity of minimizers of quasi perimeters with a volume constraint, Interfaces Free Bound. 7 (2005), no. 3, 339–352. 10.4171/IFB/128Search in Google Scholar
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