Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

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

Uniqueness and radial symmetry of minimizers for a nonlocal variational problem

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 and 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.

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