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On the existence and uniqueness of solution to a stochastic simplified liquid crystal model
Uniqueness and radial symmetry of minimizers for a nonlocal variational problem
IME-USP- Rua do Matão, 1010, CEP: 05508-090, Sao Paulo, SP, Brazil |
$ -n<p<0, $ |
$ 0<q $ |
$ K(x) = \frac{\|x\|^q}{q} - \frac{\|x\|^p}{p}, $ |
$ E(u) = \int_{R^n\times R^n} K(x-y) u(x)u(y) \,dx \,dy $ |
$ \int_{R^n}u(x) \, dx = m>0; \quad 0 \leq u(x) \leq M, $ |
$ m $ |
$ M $ |
$ -n<p<0 $ |
$ q = 2 $ |
$ -n<p<0 $ |
$ 2\leq q \leq 4. $ |
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, 1st edition, Academic Press,
1964. |
[7] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd 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, 1st edition, Academic Press,
1964. |
[7] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd edition, American Mathematical Society, 2001.
doi: 10.1090/gsm/014. |
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