Uniform holder regularity with small exponent in competition-fractional diffusion systems

Discrete and Continuous Dynamical Systems- Series A

Volumn 34, Issue 6, 2014, Pages 2669-2691

  Terracini, Susanna ^a   Verzini, Gianmaria ^b   Zilio, Alessandro ^b  

^a UNIVERSITY OF TURIN   (Italy)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Fractional laplacian; Optimal regularity of limiting profiles; Singular perturbations; Spatial segregation; Strongly competing systems

Indexed keywords

EID: 84892150488     PISSN: 10780947     EISSN: 15535231     Source Type: Journal    
DOI: 10.3934/dcds.2014.34.2669     Document Type: Article

References (21)

1. L. A. Caarelli, A. L. Karakhanyan and F.-H. Lin, The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351.
2. L. A. Caarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862.
3. L. A. Caarelli and J.-M. Roquejore, Uniform Holder estimates in a class of elliptic systems and applications to singular limits in models for diusion ames, Arch. Ration. Mech. Anal., 183 (2007), 457-487.
4. L. A. Caarelli, J.-M. Roquejore and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.
5. L. A. Caarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
6. L. A. Caarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Dierential Equations, 32 (2007), 1245-1260.
7. M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., 198 (2003), 160-196.
8. M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.
9. E. N. Dancer, K. Wang and Z. Zhang, Uniform Holder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Dierential Equations, 251 (2011), 2737-2769.
10. Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005.
11. The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.
12. E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Dierential Equations, 7 (1982), 77-116.
13. N. S. Landkof, Foundations of modern potential theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer- Verlag, New York-Heidelberg, 1972.
14. B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Holder bounds for nonlinear Schrodinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
15. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientic & Technical, Harlow, 1990.
16. X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional laplacian: regularity up to the boundary, J. Math. Pures Appl., in press.
17. L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
18. H. Tavares and S. Terracini, Regularity of the nodal set of segregated critical congurations under a weak reection law, Calc. Var. Partial Dierential Equations, 45 (2012), 273-317.
19. S. Terracini, G. Verzini and A. Zilio, Uniform Holder bounds for strongly competing systems involving the square root of the Laplacian, preprint, arXiv:1211.6087.
20. K. Wang and Z. Zhang, Some new results in competing systems with many species, Ann. Inst. H. Poincare Anal. Non Lineaire, 27 (2010), 739-761.
21. J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317.