|
|
两个相位的非局部等周问题的基态解的存在性
|
Abstract:
本文主要研究了具有幂律势的两个相位的非局部等周问题。利用集中紧性原理，证明了具有两个相位的非局部等周问题基态解的存在性。本文是在泛函的极小值的存在性的背景下，研究了两个相位的非局部等周问题的基态解的存在性，主要运用集中紧性方法，证明了紧性成立，从而证明基态解的存在性。
A nonlocal isoperimetric problem of two phases with power-law potentials is investigated. Using the concentration-compactness lemma, we prove the existence of ground states for the nonlocal isop-erimetric problem with two phases. In this paper, the existence of the ground state solution of the nonlocal isoperimetric problem with two phases is investigated in the context of the existence of the minimal value of the generalized function, and the existence of the ground state solution is proved by proving that the compactness holds, using the method of concentration-compactness lemma.
| [1] | Choksi, R., Fetecau, R.C. and Topaloglu, I. (2015) On Minimizers of Interaction Functionals with Competing Attractive and Repulsive Potentials. Annales de L’institut Henri Poincaré Analyse non Linéaire, 32, 1283-1305.
https://doi.org/10.1016/j.anihpc.2014.09.004 |
| [2] | Zhang, G. and Geng, X. (2019) On an Isoperimetric Problem with Power-Law Potentials and External Attraction. Journal of Mathematical Analysis and Applications, 482, Article ID: 123521.
https://doi.org/10.1016/j.jmaa.2019.123521 |
| [3] | Alama, S., Bronsard, L., Choksi, R. and Topaloglu, I. (2017) Ground-States for the Liquid Drop and TFDW Models with Long-Range Attraction. Journal of Mathematical Physics, 58, Article ID: 103503.
https://doi.org/10.1063/1.4999495 |
| [4] | Burchard, A., Choksi, R. and Topaloglu, I. (2018) Nolocal Shape Optimi-zation via Interactions of Attractive and Repulsive Potentials. Indiana University Mathematics Journal, 67, 375-395. https://doi.org/10.1512/iumj.2018.67.6234 |
| [5] | Maggi, F. (2012) Sets of Finite Perimeter and Geometric Varia-tional Journey. Cambridge University Press, Cambridge. |
| [6] | Frank, R.L. and Lieb, E.H. (2015) A Compactness Lem-ma and Its Application to the Existence of Minimizers for the Liquid Model. SIAM Journal on Mathematical Analysis, 47, 4436-4450. https://doi.org/10.1137/15M1010658 |
| [7] | Bonacini, M., Knüphr, H. and R？ger, M. (2016) Optimal Distribution of Oppositely Charged Phase: Perfect Screening and Other Properties. SIAM Journal on Mathematical Analysis, 48, 1128-1154. https://doi.org/10.1137/15M1020927 |
| [8] | Lu, J. and Otto, F. (2015) An Isoperimetric Problem with Coulomb Repulsion and Attraction to a Background Nucleus.
https://arxiv.org/abs/1508.07172 |
| [9] | Bonacini, M. and Cristorferi, R. (2014) Local and Global Minimality Re-sults for a Nonlocal Isoperimetric Problem. SIAM Journal on Mathematical Analysis, 46, 2310-2349. https://doi.org/10.1137/130929898 |
| [10] | Cicalese, M., Luca, L.D. and Novaga, M. (2014) Ground States of a Two Phase Model with Cross and Self Attractive Interactions. SIAM Journal on Mathematical Analysis, 48, 3412-3443. https://doi.org/10.1137/15M1033976 |
| [11] | Milbers, Z. and Schuricht, F. (2011) Some Special Aspects Related to the 1-Laplace Operator. Advances in Calculus of Variations, 4, 101-126. https://doi.org/10.1515/acv.2010.021 |
| [12] | Castro, A.D., Novaga, M., Ruffini, B. and Valdinoci, E. (2015) Non-local Quantitative Isoperimetric Inequalities. Calculus of Variations and Partial Differential Equations, 54, 2421-2464. https://doi.org/10.1007/s00526-015-0870-x |
| [13] | Chang, K.C. (2009) The Spectrum of the 1-Laplace Operator. Communications in Contemporary Mathematics, 11, 865-894. https://doi.org/10.1142/S0219199709003570 |
| [14] | Carlen, E.A., Carvalho, M., Esposito, R., Lebowitz, J.L. and Marra, R. (2003) Free Energy Minimizers for a Two-Species Model with Segregation and Liquid-Vapor Transition. Nonlinearity, 16, 1075-1105.
https://doi.org/10.1088/0951-7715/16/3/316 |
