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

Indiana University Mathematics Journal

Volumn 54, Issue 3, 2005, Pages 779-815

  Conti, Monica ^a   Terracini, Susanna ^b   Verzini, Gianmaria ^a  

^a POLITECNICO DI MILANO   (Italy)

^b UNIVERSITY OF MILANO BICOCCA   (Italy)

Author keywords

Monotonicity formula; Multiple intersection points; Regularity theory; Segregation states

Indexed keywords

EID: 23244445487     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2005.54.2506     Document Type: Article

References (31)

1. GIOVANNI ALESSANDRINI, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 229-256 (1988). MR 89e:35040
2. HANS WILHELM ALT, LUIS A. CAFFARELLI, and AVNER FRIEDMAN, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431-461. MR 85h:49014
3. I. ATHANASOPOULOS, LUIS A. CAFFARELLI, and S. SALSA, The free boundary in an inverse conductivity problem, J. Reine Angew. Math. 534 (2001), 1-31. MR2002d:35219
4. J. BLAT and K.J. BROWN, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21-34. MR 85k:92053
5. LUIS A. CAFFARELLI, LAVI KARP, and HENRIK SHAHGHOLIAN, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. (2) 151 (2000), 269-292. MR 2001a:35188
6. LUIS A. CAFFARELLI, DAVID JERISON, and CARLOS E. KENIG, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. (2) 155 (2002), 369-404. MR 2003B5068
7. _, Regularity for inhomogeneous two-phase free boundary problems (in preparation).
8. M. CONTI, S. TERRACINI, and G. VERZINS Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 871-888, http://dx.doi.org/10.1016/50294-1449(02)00104-X. MR 2003i:35084 (English, with English and French summaries)
9. _, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), 160-196, http://dx.doi.org/10.1016/50022-1236(02)00105- 2. MR 2004h:35171
10. CHRIS COSNER and A.C. LAZER, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math. 44 (1984), 1112-1132. MR 86j:92020
11. E.N. DANCER, Competing species systems with diffusion and large interactions, Rend. Sem. Mat. Fis. Milano 65 (1995), 23-33 (1997). MR 99d:92023
12. E.N. DANCER and YI HONG DU, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), 434-475. MR 95i:35296
13. _, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal. 24 (1995), 337-357, http://dx.doi.org/10.1016/0362-546X(94)E0063-M. MR 96c:92028
14. _, Positive solutions for a three-species competition system with diffusion. II. The case of equal birth rates, Nonlinear Anal. 24 (1995), 359-373, http://dx.doi.org/10.1016/0362-546X(94)E0064-N. MR 96c:92029
15. _, On a free boundary problem arising from population biology, Indiana Univ. Math. J. 52 (2003), 51-67. MR2004c:35423
16. E.N. DANCER and Z.M. GUO, Uniqueness and stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlinear Anal. 1 (1994), 19-45. MR95i:35159
17. E.N. DANCER, D. HILHORST, M. MIMURA, and L.A. PELZTIER, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math. 10 (1999), 97-115, http://dx.doi.org/10.1017/S0956792598003660. MR 2001a:92030
18. S.-I. EI and E. YANAGIDA, Dynamics of interfaces in competition-diffusion systems, SIAM J. Appl. Math. 54 (1994), 1355-1373. MR95B5109
19. J.C. EILBECK, J.E. FURTER, and J. LÓPEZ-GÓMEZ, Coexistence in the competition model with diffusion, J. Differential Equations 107 (1994), 96-139, http://dx.doi.org/10.1006/jdeq.1994.1005. MR 95c:35072
20. LAWRENCE C. EVANS, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998, ISBN 0-8218-0772-2. MR 99e:35001
21. CHANGFENG GUI and YUAN LOU, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math. 47 (1994), 1571-1594. MR 95k:35108
22. PHILIP HARTMAN and AUREL WINTNER, On the local behavior of solutions of non-parabolic partial differential equations. III. Approximations by spherical harmonics, Amer. J. Math. 77 (1955), 453-474. MR 17,855a
23. PHILIP KORMAN and ANTHONY LEUNG, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26 (1987), 145-160. MR 89e:92045
24. A.C. LAZER and P.J. MCKENNA, On steady state solutions of a system of reaction-diffusion equations from biology, Nonlinear Anal. 6 (1982), 523-530, http://dx.doi.org/10.1016/0362-546X(82)90045-1. MR 83h:35014
25. YUAN LOU, SALOMÉ MARTÍNEZ, and WEI-MING NI, On 3 × 3 Lotka-Volterra competition systems with cross-diffusion, Discrete Contin. Dynam. Systems 6 (2000), 175-190. MR 2001c:92014
26. YUAN LOU and WEI-MING NI, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996), 79-131, http://dx.doi.org/10.1006/jdeq. 1996.0157. MR97i:35086
27. HIROSHI MATANO and MASAYASU MIMURA, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci. 19 (1983), 1049-1079. MR 85e:35065
28. MASAYASU MIMURA, Spatial distribution of competing species, Mathematical Ecology (Trieste, 1982), Lecture Notes in Biomath., vol. 54, Springer, Berlin, 1984, pp. 492-501. 760 308
29. NANAKO SHIGESADA, KOHKICHI KAWASAKI, and EI TERAMOTO, The effects of interference competition on stability, structure and invasion of a multispecies system, J. Math. Biol. 21 (1984), 97-113. MR 86d:92025
30. MICHAEL STRUWE, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34, Springer-Verlag, Berlin, 2000, ISBN 3-540-66479-3. MR2000i:49001
31. YOSHIHIRO TONEGAWA, On the regularity of a chemical reaction interface, Comm. Partial Differential Equations 23 (1998), 1181-1207. MR99i:35058