By Conti M., Terracini S., Verzini G.
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Extra info for A variational problem for the spatial segregation of reaction-diffusion systems
155 (2002), no. A. Caffarelli, D. E. Kenig, Regularity for inhommogeneous two–phase free boundary problems, in preparation  M. Conti, S. Terracini, G. Verzini Nehari’s Problem and Competing Species Systems, Ann. Inst. H. Poincar´e, AN 19, 6 (2002) 871–888  M. Conti, S. Terracini, G. Verzini An optimal partition problem related to nonlinear eigenvalues, Journal of Funct. Anal , to appear  C. Cosner, A. Lazer, Stable coexistence in the Volterra–Lotka competition model with diffusion, SIAM J.
Du, Positive solutions for a three-species competition system with diffusion. II. N. H. Du, Positive solutions for a three-species competition system with diffusion. I. N. M. Guo, Uniqueness and stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlinear Anal. N. Dancer, D. Hilhorst, M. A. Peletier, Spatial segregation limit of a competition– diffusion system, European J. Appl. Math. 10 (1999), 97–115  S-I. Ei, E. Yanagida, Dynamics of interfaces in competition-diffusion systems, SIAM J.
1, we argue by induction over the number h of connected components of the set Z3 . 8 there is at most one minimal loop of the adjacency relation. 8 gives the desired assertion. 7. Now, let the Theorem be true for h and assume that Z3 has h + 1 connected components. Again, if the adjacency relation has no loops we are done. 8 to treat those connected components contained in the interior of the minimal loop and the inductive hypothesis to treat all those contained in the outer region. 1 Having proved that the multiple points are isolated, the existence of points of multiplicity zero can be easily ruled out for connected domains.