Existence of Ground States of Nonlocal-Interaction Energies
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We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. \(H\)-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.
KeywordsGround states Global minimizers H-stability Pair potentials Self-assembly Aggregation
Mathematics Subject Classification49J45 82B21 82B05 35R09 45K05
The authors are thankful to Rustum Choksi for initiating this collaboration and to José Antonio Carrillo for a valuable discussion. The authors would like to thank the Center for Nonlinear Analysis of the Carnegie Mellon University for its support, and hospitality during IT’s visit. RS was supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Carnegie Mellon Portugal Program under Grant SFRH/BD/33778/2009. DS is grateful to NSF (Grant DMS-1211760) and FCT (Grant UTA CMU/MAT/0007/2009). IT was also partially supported by the Applied Mathematics Laboratory of the Centre de Recherches Mathématiques. The research was also supported by NSF PIRE Grant OISE-0967140.
- 1.Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differ. Equ. 44(1–2), 81–100 (2012)Google Scholar
- 4.Balagué, D., Carrillo, J.A., Yao, Y.: Confinement for attractive–repulsive kernels. Discret. Contin. Dyn. S. (to appear)Google Scholar
- 12.J.A. Cañizo, Carrillo, J.A. and Patacchini F.S.: Existence of compactly supported global minimizers for the interaction energy. preprint, 2014Google Scholar
- 14.Carrillo, J.A., Chipot, M., and Huang Y.: On global minimizers of repulsive-attractive power-law interaction energies. preprint, 2014Google Scholar
- 15.Carrillo, J.A., Delgadino, M.G., and Mellet, A.: Regularity of local minimizers of the interaction energy via obstacle problems. preprint, 2014Google Scholar
- 19.Choksi, R., Fetecau, R.C., and Topaloglu, I.: On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire.( to appear)Google Scholar
- 25.Haile, J.M.: Molecular Dynamics Simulation: Elementary Methods. Wiley, New York (1992)Google Scholar
- 32.Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1(2):109–145 (1984)Google Scholar