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

Previous Article
On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains
DCDS Home
This Issue
Next Article

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
Related Papers

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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s11784-009-0110-0. Google Scholar
[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. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar
[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. doi: 10.1007/s00205-006-0013-9. Google Scholar
[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. doi: 10.4171/JEMS/226. Google Scholar
[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. doi: 10.1007/s00222-007-0086-6. Google Scholar
[6]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar
[7]	M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues,, J. Funct. Anal., 198 (2003), 160. doi: 10.1016/S0022-1236(02)00105-2. Google Scholar
[8]	M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems,, Adv. Math., 195 (2005), 524. doi: 10.1016/j.aim.2004.08.006. Google Scholar
[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. doi: 10.1016/j.jde.2011.06.015. Google Scholar
[10]	________, Dynamics of strongly competing systems with many species,, Trans. Amer. Math. Soc., 364 (2012), 961. doi: 10.1090/S0002-9947-2011-05488-7. Google Scholar
[11]	_______, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, J. Funct. Anal., 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013. Google Scholar
[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. doi: 10.1080/03605308208820218. Google Scholar
[13]	N. S. Landkof, Foundations of modern potential theory,, Translated from the Russian by A. P. Doohovskoy, (1972). Google Scholar
[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. doi: 10.1002/cpa.20309. Google Scholar
[15]	B. Opic and A. Kufner, Hardy-type inequalities,, Pitman Research Notes in Mathematics Series, (1990). Google Scholar
[16]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional laplacian: regularity up to the boundary,, J. Math. Pures Appl., (). doi: 10.1016/j.matpur.2013.06.003. Google Scholar
[17]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar
[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. doi: 10.1007/s00526-011-0458-z. Google Scholar
[19]	S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian,, preprint, (). Google Scholar
[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. doi: 10.1016/j.anihpc.2009.11.004. Google Scholar
[21]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar

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. doi: 10.1007/s11784-009-0110-0. Google Scholar
[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. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar
[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. doi: 10.1007/s00205-006-0013-9. Google Scholar
[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. doi: 10.4171/JEMS/226. Google Scholar
[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. doi: 10.1007/s00222-007-0086-6. Google Scholar
[6]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar
[7]	M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues,, J. Funct. Anal., 198 (2003), 160. doi: 10.1016/S0022-1236(02)00105-2. Google Scholar
[8]	M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems,, Adv. Math., 195 (2005), 524. doi: 10.1016/j.aim.2004.08.006. Google Scholar
[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. doi: 10.1016/j.jde.2011.06.015. Google Scholar
[10]	________, Dynamics of strongly competing systems with many species,, Trans. Amer. Math. Soc., 364 (2012), 961. doi: 10.1090/S0002-9947-2011-05488-7. Google Scholar
[11]	_______, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, J. Funct. Anal., 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013. Google Scholar
[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. doi: 10.1080/03605308208820218. Google Scholar
[13]	N. S. Landkof, Foundations of modern potential theory,, Translated from the Russian by A. P. Doohovskoy, (1972). Google Scholar
[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. doi: 10.1002/cpa.20309. Google Scholar
[15]	B. Opic and A. Kufner, Hardy-type inequalities,, Pitman Research Notes in Mathematics Series, (1990). Google Scholar
[16]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional laplacian: regularity up to the boundary,, J. Math. Pures Appl., (). doi: 10.1016/j.matpur.2013.06.003. Google Scholar
[17]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar
[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. doi: 10.1007/s00526-011-0458-z. Google Scholar
[19]	S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian,, preprint, (). Google Scholar
[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. doi: 10.1016/j.anihpc.2009.11.004. Google Scholar
[21]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar

[1]	Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025
[2]	De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431
[3]	Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
[4]	Eun Kyoung Lee, R. Shivaji, Jinglong Ye. Classes of singular $pq-$Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1123-1132. doi: 10.3934/dcds.2010.27.1123
[5]	Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576
[6]	Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[7]	Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[8]	Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
[9]	Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
[10]	Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617
[11]	King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051
[12]	Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245
[13]	Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-29. doi: 10.3934/dcdsb.2019170
[14]	Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981
[15]	Ami B. Shah, Katarzyna A. Rejniak, Jana L. Gevertz. Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1185-1206. doi: 10.3934/mbe.2016038
[16]	Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141
[17]	Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335
[18]	Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[19]	Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065
[20]	Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences