A variational problem for the spatial segregation of reaction-diffusion systems

By Monica Conti, S. Terracini, Gianmaria Verzini. 2003

166 citations recorded by the source corpus.

In this paper we study a class of stationary states for reaction-diffusion systems of k ≥ 3 densities having disjoint supports. For a class of segregation states governed by a variational principle we prove existence and provide conditions for uniqueness. Some qualitative properties and the local regularity both of the densities and of their free boundaries are established in the more general context of a functional class characterized by differential inequalities.

Subfields: Applied Mathematics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/662c802f8475160afb4799c8b3b08fb051f48be3

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In Applied Mathematics

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Citations in the corpus

The geometry of solutions to a segregation problem for nondivergence systems (2009, background): “This problem has received a considerable attention in particular [CTV][CTV2] [CTV3][ CTV4 ] vari-”
Dynamics of strongly competing systems with many species (2012, unknown): “Conti, Terracini and Verzini [9], [10], Caffarelli, Karakhanyan and Lin [2] [3], etc.”
Some new results in competing systems with many species (2010, methodology): “Conti, Terracini and Verzini [5,6], Caffarelli, Karakhanyan and Lin [1,2], etc., established the regularity of the singular limit (and the partial regularity of its free boundary) as κ → +∞ and the uniform regularity for all κ > 0.”
Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics (2008, background): “For the last two decades there has been tremendous interest, by both mathematicians and ecologists, in two-species Lotka-Volterra competition models with spatially homogeneous or heterogeneous interactions, see [9, 10, 11, 12, 18, 19, 28, 29, 33, 35, 38, 39, 40, 41, 46, 55, 59, 62, 63, 64, 65, 66, 7”
On Spectral Minimal Partitions: A Survey (2010, methodology): “It has been proved by Conti-Terracini-Verzini [CTV1, CTV2 , CTV3] (existence) and Helffer–Hoffmann-Ostenhof–Terracini [HHOT1] (regularity) the following theorem:”
On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae (2003, methodology): “problems arise in connection with the phenomenon of the spatial segregation in reaction–diffusion systems, as shown in [ 8 ,9].”
Nodal Domains and Spectral Minimal Partitions (2006, methodology): “It has been proved1 by Conti-Terracini-Verzini [7, 8, 9] that Theorem 1.11 For any k, there exists a k– minimal regular strong partition.”
Asymptotic estimates for the spatial segregation of competitive systems (2005, methodology): “In the recent paper [ 10 ], the authors have developed a theory concerning the regularity and the properties of the free boundary of the elements of analogous functional classes.”