Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

Previous Article
On the existence and uniqueness of solution to a stochastic simplified liquid crystal model
CPAA Home
This Issue
Next Article
Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines

September 2019, 18(5): 2265-2282. doi: 10.3934/cpaa.2019102

Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

Orlando Lopes

IME-USP- Rua do Matão, 1010, CEP: 05508-090, Sao Paulo, SP, Brazil

Received August 2017 Revised November 2017 Published April 2019

For

$ -n<p<0, $

$ 0<q $

and

$ K(x) = \frac{\|x\|^q}{q} - \frac{\|x\|^p}{p}, $

the existence of minimizers of

$ E(u) = \int_{R^n\times R^n} K(x-y) u(x)u(y) \,dx \,dy $

under

$ \int_{R^n}u(x) \, dx = m>0; \quad 0 \leq u(x) \leq M, $

with given

$ m $

and

$ M $

, is proved in [3]. Moreover, except for translation, uniqueness and radial symmetry of the minimizer is proved for

$ -n<p<0 $

and

$ q = 2 $

. Here in the present paper, we show that, except for translation, uniqueness and radial symmetry of the minimizer hold for

$ -n<p<0 $

and

$ 2\leq q \leq 4. $

Applications are given.

Keywords: Radial symmetry, nonlocal variational problems, uniqueness of minimizers, shape of minimizers, phases of minimizers.

Mathematics Subject Classification: 34A34.

Citation: Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102

References:

[1]	A. Burchard, R. Choksi and I. Topaloglu, Nonlocal shape optimization via interactions of attractive and repulsive potentials, Indiana Univ. Math. J., to appear. doi: 10.1512/iumj.2018.67.6234.
[2]	J. A. Cañizo, J. A. Carrillo and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Archive for Rational Mechanics and Analysis, 217 (2015), 1197-1217. doi: 10.1007/s00205-015-0852-3.
[3]	R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 1283-1305. doi: 10.1016/j.anihpc.2014.09.004.
[4]	R. L. Frank and E. H. Lieb, A 'liquid-solid' phase transition in a simple model for swarming, based on the 'no flat-spots' theorem for subharmonic functions, Indiana University Mathematical Journal, to appear. doi: 10.1512/iumj.2018.67.7398.
[5]	R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716. doi: 10.1088/0951-7715/24/10/002.
[6]	I. M. Gelfand and G. E. Shilov, Generalized Functions, vol. 1, 1^st edition, Academic Press, 1964.
[7]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2^nd edition, American Mathematical Society, 2001. doi: 10.1090/gsm/014.

show all references

References:

[1]	A. Burchard, R. Choksi and I. Topaloglu, Nonlocal shape optimization via interactions of attractive and repulsive potentials, Indiana Univ. Math. J., to appear. doi: 10.1512/iumj.2018.67.6234.
[2]	J. A. Cañizo, J. A. Carrillo and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Archive for Rational Mechanics and Analysis, 217 (2015), 1197-1217. doi: 10.1007/s00205-015-0852-3.
[3]	R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 1283-1305. doi: 10.1016/j.anihpc.2014.09.004.
[4]	R. L. Frank and E. H. Lieb, A 'liquid-solid' phase transition in a simple model for swarming, based on the 'no flat-spots' theorem for subharmonic functions, Indiana University Mathematical Journal, to appear. doi: 10.1512/iumj.2018.67.7398.
[5]	R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716. doi: 10.1088/0951-7715/24/10/002.
[6]	I. M. Gelfand and G. E. Shilov, Generalized Functions, vol. 1, 1^st edition, Academic Press, 1964.
[7]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2^nd edition, American Mathematical Society, 2001. doi: 10.1090/gsm/014.

[1]	Christos Gavriel, Richard Vinter. Regularity of minimizers for second order variational problems in one independent variable. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 547-557. doi: 10.3934/dcds.2011.29.547
[2]	Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411
[3]	P. Di Gironimo, L. D’Onofrio. On the regularity of minimizers to degenerate functionals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1311-1318. doi: 10.3934/cpaa.2010.9.1311
[4]	Armands Gritsans, Felix Sadyrbaev. The Nehari solutions and asymmetric minimizers. Conference Publications, 2015, 2015 (special) : 562-568. doi: 10.3934/proc.2015.0562
[5]	Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure & Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341
[6]	Nicholas D. Alikakos, Giorgio Fusco. Density estimates for vector minimizers and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5631-5663. doi: 10.3934/dcds.2015.35.5631
[7]	Giovanni Bellettini, Matteo Novaga, Shokhrukh Yusufovich Kholmatov. Minimizers of anisotropic perimeters with cylindrical norms. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1427-1454. doi: 10.3934/cpaa.2017068
[8]	Menita Carozza, Gioconda Moscariello, Antonia Passarelli. Higher integrability for minimizers of anisotropic functionals. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 43-55. doi: 10.3934/dcdsb.2009.11.43
[9]	Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems & Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012
[10]	Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687
[11]	Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1325-1346. doi: 10.3934/dcds.2011.31.1325
[12]	Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
[13]	Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
[14]	Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure & Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415
[15]	Giorgio Fusco. On some elementary properties of vector minimizers of the Allen-Cahn energy. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1045-1060. doi: 10.3934/cpaa.2014.13.1045
[16]	Duvan Henao, Rémy Rodiac. On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4509-4536. doi: 10.3934/dcds.2018197
[17]	Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 251-265. doi: 10.3934/dcdss.2019018
[18]	Vladimir Bobkov, Mieko Tanaka. Remarks on minimizers for (p, q)-Laplace equations with two parameters. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1219-1253. doi: 10.3934/cpaa.2018059
[19]	Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control & Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399
[20]	Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327

American Institute of Mathematical Sciences