A "liquid-solid" phase transition in a simple model for swarming, based on the "no flat-spots" theorem for subharmonic functions

DOI10.1512/iumj.2018.67.7398zbMath1420.49040arXiv1607.07971OpenAlexW2620806331WikidataQ129322695 ScholiaQ129322695MaRDI QIDQ4962409

Elliott H. Lieb, Rupert L. Frank

Publication date: 2 November 2018

Published in: Indiana University Mathematics Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1607.07971

zbMATH Keywords

shape optimizationstrictly subharmonic function

Mathematics Subject Classification ID

Regularity of solutions in optimal control (49N60) Harmonic, subharmonic, superharmonic functions in higher dimensions (31B05) Optimization of shapes other than minimal surfaces (49Q10)

Related Items (18)

Primal dual methods for Wasserstein gradient flows ⋮ Geometrical Bounds for Variance and Recentered Moments ⋮ Classifying Minimum Energy States for Interacting Particles: Spherical Shells ⋮ On the strong attraction limit for a class of nonlocal interaction energies ⋮ Some minimization problems for mean field models with competing forces ⋮ Stability of the Laughlin phase against long-range interactions ⋮ Equilibria and energy minimizers for an interaction model on the hyperbolic space ⋮ Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits ⋮ Local incompressibility estimates for the Laughlin phase ⋮ Proof of spherical flocking based on quantitative rearrangement inequalities ⋮ Critical weak-\(L^p\) differentiability of singular integrals ⋮ Isodiametry, variance, and regular simplices from particle interactions ⋮ Aggregation-diffusion to constrained interaction: minimizers \& gradient flows in the slow diffusion limit ⋮ Minimizers for a one-dimensional interaction energy ⋮ Uniqueness and radial symmetry of minimizers for a nonlocal variational problem ⋮ Singular support of minimizers of the causal variational principle on the sphere ⋮ Least Wasserstein distance between disjoint shapes with perimeter regularization ⋮ From radial symmetry to fractal behavior of aggregation equilibria for repulsive-attractive potentials

Cites Work

• Regularity of local minimizers of the interaction energy via obstacle problems
• On minimizers of interaction functionals with competing attractive and repulsive potentials
• Elliptic partial differential equations of second order
• Existence of ground states of nonlocal-interaction energies
• Existence of compactly supported global minimisers for the interaction energy
• Equilibria of biological aggregations with nonlocal repulsive-attractive interactions
• Absolute continuity on tracks and mappings of Sobolev spaces
• Partial regularity of solutions of fully nonlinear, uniformly elliptic equations
• Nonexistence of a Minimizer for Thomas-Fermi-Dirac-von Weizsäcker Model
• A Primer of Swarm Equilibria
• Swarm dynamics and equilibria for a nonlocal aggregation model
• Symmetric Decreasing Rearrangement Is Sometimes Continuous
• Hahn decomposition for the Riesz charge of $\delta$-subharmonic functions
• Nonlocal Shape Optimization via Interactions of Attractive and Repulsive Potentials
• A General Chain Rule for Derivatives and the Change of Variables Formula for the Lebesgue Integral
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