Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

2014/07/31 by Nicola Soave, Alessandro Zilio · 69 citations
Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Artificial intelligence #Complex system #Computer science #Lipschitz continuity #Lipschitz domain #Mathematics #Nonlinear system #Physics #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math.AP #msc:35B25 #msc:35B65 #msc:35J47 #msc:35R35 #msc:81Q05 #msc:92D25

paper · pdf · doi:10.1007/s00205-015-0867-9

published in Archive for Rational Mechanics and Analysis 218(2), 647-697 (Springer Science+Business Media) · to appear on Archive for Rational Mechanics and Analysis

arxiv created 2015/04/02 · openalex publication_date 2015/04/10 · arxiv updated 2016/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

For a class of systems of semi-linear elliptic equations, including -Δu_i=f_i(x,u_i) - βu_i∑_{j≠ i}a_iju_j^p, i=1,…,k, for p=2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L^∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→ +∞, that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proof rests on monotonicity formulae of Alt-Caffarelli-Friedman and Almgren type in the variational setting and Caffarelli-Jerison-Kenig in the symmetric one.

Citations

• Some New Monotonicity Theorems with Applications to Free Boundary Problems
• A Free Boundary Problem Arising from Segregation of Populations with High Competition
• On Phase-Separation Models: Asymptotics and Qualitative Properties
• Uniform Hölder Estimates in a Class of Elliptic Systems and Applications to Singular Limits in Models for Diffusion Flames
• Variational problems with two phases and their free boundaries

Cited by

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• A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems
• Generic configurations in 2D strongly competing systems
• Competition in periodic media: III -- Existence & stability of segregated periodic coexistence states
• On the emergence of dead cores in elliptic systems with sublinear competitive interactions
• A survey on the optimal partition problem
• Spectral minimal partitions of unbounded domains
• Predator-prey models with competition, Part II: uniform regularity estimates
• Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation
• Predator-prey models with competition, Part III: Classification of stationary solutions
• Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem
• Bifurcation and segregation in quadratic two-populations Mean Field Games systems
• Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues
• On a two-component Bose-Einstein condensate with steep potential wells

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