Strong Competition versus Fractional Diffusion: The Case of Lotka-Volterra Interaction

Communications in Partial Differential Equations

Volumn 39, Issue 12, 2014, Pages 2284-2313

  Verzini, Gianmaria ^a   Zilio, Alessandro ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

Blow up analysis; Monotonicity formulae; Optimal regularity of limiting profiles; Singular perturbations; Spatial segregation

Indexed keywords

EID: 84907584921     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605302.2014.890627     Document Type: Article

References (30)

1. Alt , H.W. , Caffarelli , L.A. , Friedman , A. (1984). Variational problems with two phases and their free boundaries . Trans. Amer. Math. Soc. 282 : 431 – 461 .
2. Athanasopoulos , I. , Caffarelli , L.A. (2006). Optimal regularity of lower-dimensional obstacle problems . Journal of Mathematical Sciences 132 : 274 – 284 .
3. Bonforte , M. , Vázquez , J.L. (2014). Quantitative local and global a priori estimates for fractional nonlinear diffusion equations . Adv. Math. 250 : 242 – 284 .
4. Cabré , X. , Sire , Y. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates . Ann. Inst. H. Poincaré Anal. Non Linéaire. In press.
5. Caffarelli , L.A. , Jerison , D. , Kenig , C.E. (2002). Some new monotonicity theorems with applications to free boundary problems . Ann. of Math. (2) 155 : 369 – 404 .
6. Caffarelli , L.A., Karakhanyan , A.L., Lin , F.-H. (2009). The geometry of solutions to a segregation problem for nondivergence systems. J. Fixed Point Theory Appl.5:319–351.
7. Caffarelli , L.A. , Lin , F.-H. (2008). Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries . J. Amer. Math. Soc. 21 : 847 – 862 .
8. Caffarelli , L.A. , Roquejoffre , J.-M. , Sire , Y. (2010). Variational problems for free boundaries for the fractional Laplacian . J. Eur. Math. Soc. (JEMS) 12 : 1151 – 1179 .
9. Caffarelli , L.A. , Salsa , S. (2005). A Geometric Approach to Free Boundary Problems . Graduate Studies in Mathematics , Vol. 68 . Providence , RI : American Mathematical Society .
10. Caffarelli , L.A. , Salsa , S. , Silvestre , L. (2008). Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian . Invent. Math. 171 : 425 – 461 .
11. Caffarelli , L.A. , Silvestre , L. (2007). An extension problem related to the fractional Laplacian . Comm. Partial Differential Equations 32 : 1245 – 1260 .
12. Chang , S.-M. , Lin , C.-S. , Lin , T.-C. , Lin , W.-W. (2004). Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates . Phys. D. 196 (3–4): 341 – 361 .
13. Conti , M. , Terracini , S. , Verzini , G. (2002). Nehari's problem and competing species systems . Ann. Inst. H. Poincaré Anal. Non Linéaire 19 : 871 – 888 .
14. Conti , M. , Terracini , S. , Verzini , G. (2003). An optimal partition problem related to nonlinear eigenvalues . J. Funct. Anal. 198 : 160 – 196 .
15. Conti , M. , Terracini , S. , Verzini , G. (2005). Asymptotic estimates for the spatial segregation of competitive systems . Adv. Math. 195 : 524 – 560 .
16. Conti , M. , Terracini , S. , Verzini , G. (2005). A variational problem for the spatial segregation of reaction-diffusion systems . Indiana Univ. Math. J. 54 : 779 – 815 .
17. Dancer , E. N. , Wang , K. , Zhang , Z. (2011). Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species . J. Diff. Eqs. 251 : 2737 – 2769 .
18. Dancer , E.N. , Wang , K. , Zhang , Z. (2012). The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture . J. Funct. Anal. 262 : 1087 – 1131 .
19. de Pablo , A. , Quirós , F. , Rodríguez , A. , Vázquez , J.L. (2012). A general fractional porous medium equation . Comm. Pure Appl. Math. 65 : 1242 – 1284 .
20. Dipierro , S. , Pinamonti , A. (2013). A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian . J. Diff. Eqs. 255 : 85 – 119 .
21. Fall , M. M. , Felli , V. (2014). Unique continuation property and local asymptotics of solutions to fractional elliptic equations . Comm. Part. Diff. Eqs. 39 : 354 – 397 .
22. Jin , T. , Li , Y. , Xiong , J. On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. Preprint, arXiv:1111.1332.
23. Noris , B. , Tavares , H. , Terracini , S. , Verzini , G. (2010). Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition . Comm. Pure Appl. Math. 63 : 267 – 302 .
24. Noris , B., Tavares , H., Terracini , S., Verzini , G. (2012). Convergence of minimax structures and continuation of critical points for singularly perturbed systems. J. Eur. Math. Soc. (JEMS)14:1245–1273.
25. Opic , B. , Kufner , A. (1990). Hardy-type Inequalities . Pitman Research Notes in Mathematics Series, Vol. 219. Harlow , UK : Longman Scientific & Technical .
26. Silvestre , L. (2007). Regularity of the obstacle problem for a fractional power of the Laplace operator . Comm. Pure Appl. Math. 60 : 67 – 112 .
27. Tavares , H. , Terracini , S. (2012). Regularity of the nodal set of segregated critical configurations under a weak reflection law . Calc. Var. Part. Diff. Eqs. 45 : 273 – 317 .
28. Terracini , S. , Verzini , G. , Zilio , A. (2014). Uniform Hölder regularity with small exponent in competition-fractional diffusion systems . Discrete Contin. Dyn. Syst. 34 : 2669 – 2691 .
29. Terracini , S. , Verzini , G. , Zilio , A. Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian. Preprint, arXiv:1211.6087.
30. Zilio , A. (2014). On monotonicity formulæ, fractional operators and strong competition. PhD thesis, Politecnico di Milano, Milano, Italy .