Some minimization problems for mean field models with competing forces

2021/09/06 by Rupert L. Frank, Frank, Rupert L. · 2 citations
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph) #Micro and Nano Robotics

paper · pdf · doi:10.48550/arxiv.2109.02393

openalex publication_date 2021/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We review recent results on three families of minimization problems, defined on subsets of nonnegative functions with fixed integral. The competition between attractive and repulsive forces leads to transitions between parameter regimes, where minimizers exist and where they do not. The problems considered are generalized liquid drop models, swarming models and generalized Keller-Segel models.

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• Continuity of critical points for 1-dimensional non-local energies
• Particle approximation of nonlocal interaction energies

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