By Conti M., Terracini S., Verzini G.
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Extra resources for A variational problem for the spatial segregation of reaction-diffusion systems
Shahgholian, Regularity of a free boundary with applications to the Pompeiu problem, Ann. of Math. A. Caffarelli, D. E. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. 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.
Then by  and  we know that only a finite number of points of Z3 lie in Ωγ . On the other hand, by the minimality of γ, Z3 ∩ Ωγ is connected. Thus the origin is the only one multiple point contained in Ωγ . 3 The set Z3 consists of isolated points. 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.
Differential Equations, 131 (1996) 1, 79–131  H. Matano, M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. , 19 (1983) 3, 1049–1079  M. Mimura, Masayasu, Spatial distribution of competing species, Mathematical ecology (Trieste, 1982), Lecture Notes in Biomath. 54, 492–501  N. Shigesada, K. Kawasaki, E. Teramoto, The effects of interference competition on stability, structure and invasion of a multispecies system, J. Math. , 21 (1984) 2, 97–113  Y.