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

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June 2014, 34(6): 2669-2691. doi: 10.3934/dcds.2014.34.2669

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

Susanna Terracini ^1, , Gianmaria Verzini ^2, and Alessandro Zilio ^2,

1.	Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino
2.	Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano, Italy

Received March 2013 Revised June 2013 Published December 2013

Abstract
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For a class of competition-diffusion nonlinear systems involving the $s$-power of the laplacian, $s\in(0,1)$, of the form \[ (-\Delta)^{s} u_i=f_i(u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that $L^\infty$ boundedness implies $\mathcal{C}^{0,\alpha}$ boundedness for $\alpha>0$ sufficiently small, uniformly as $\beta\to +\infty$. This extends to the case $s\neq1/2$ part of the results obtained by the authors in the previous paper [arXiv: 1211.6087v1].

Keywords: strongly competing systems, optimal regularity of limiting profiles, Fractional laplacian, singular perturbations., spatial segregation.

Mathematics Subject Classification: Primary: 35J65; Secondary: 35B40, 35B44, 35R11, 81Q05, 82B1.

Citation: Susanna Terracini, Gianmaria Verzini, Alessandro Zilio. Uniform Hölder regularity with small exponent in competition-fractional diffusion systems. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2669-2691. doi: 10.3934/dcds.2014.34.2669

References:

[1]	L. A. Caffarelli, 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. doi: 10.1007/s11784-009-0110-0.
[2]	L. A. Caffarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.
[3]	L. A. Caffarelli and J.-M. Roquejoffre, Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal., 183 (2007), 457-487. doi: 10.1007/s00205-006-0013-9.
[4]	L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.
[5]	L. A. Caffarelli, 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. doi: 10.1007/s00222-007-0086-6.
[6]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[7]	M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., 198 (2003), 160-196. doi: 10.1016/S0022-1236(02)00105-2.
[8]	M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006.
[9]	E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015.
[10]	________, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7.
[11]	_______, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131. doi: 10.1016/j.jfa.2011.10.013.
[12]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[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 Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309.
[15]	B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific & 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. doi: 10.1016/j.matpur.2013.06.003.
[17]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[18]	H. Tavares and S. Terracini, Regularity of the nodal set of segregated critical configurations under a weak reflection law, Calc. Var. Partial Differential Equations, 45 (2012), 273-317. doi: 10.1007/s00526-011-0458-z.
[19]	S. Terracini, G. Verzini and A. Zilio, Uniform Hölder 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. Poincaré Anal. Non Linéaire, 27 (2010), 739-761. doi: 10.1016/j.anihpc.2009.11.004.
[21]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

show all references

References:

[1]	L. A. Caffarelli, 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. doi: 10.1007/s11784-009-0110-0.
[2]	L. A. Caffarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.
[3]	L. A. Caffarelli and J.-M. Roquejoffre, Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal., 183 (2007), 457-487. doi: 10.1007/s00205-006-0013-9.
[4]	L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.
[5]	L. A. Caffarelli, 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. doi: 10.1007/s00222-007-0086-6.
[6]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[7]	M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., 198 (2003), 160-196. doi: 10.1016/S0022-1236(02)00105-2.
[8]	M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006.
[9]	E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015.
[10]	________, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7.
[11]	_______, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131. doi: 10.1016/j.jfa.2011.10.013.
[12]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[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 Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. doi: 10.1002/cpa.20309.
[15]	B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific & 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. doi: 10.1016/j.matpur.2013.06.003.
[17]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[18]	H. Tavares and S. Terracini, Regularity of the nodal set of segregated critical configurations under a weak reflection law, Calc. Var. Partial Differential Equations, 45 (2012), 273-317. doi: 10.1007/s00526-011-0458-z.
[19]	S. Terracini, G. Verzini and A. Zilio, Uniform Hölder 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. Poincaré Anal. Non Linéaire, 27 (2010), 739-761. doi: 10.1016/j.anihpc.2009.11.004.
[21]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

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