# American Institute of Mathematical Sciences

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

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

 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

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].
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., ().  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, (). [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., ().  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, (). [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.
 [1] Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure and 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 and Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [3] 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 [4] Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173 [5] Eun Kyoung Lee, R. Shivaji, Jinglong Ye. Classes of singular $pq-$Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1123-1132. doi: 10.3934/dcds.2010.27.1123 [6] Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617 [7] King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051 [8] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [9] Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 [10] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [11] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [12] Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022 [13] Dingshi Li, Xiaohu Wang, Junyilang Zhao. Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2751-2776. doi: 10.3934/cpaa.2020120 [14] Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245 [15] Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759 [16] Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044 [17] Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981 [18] Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083 [19] Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170 [20] 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

2020 Impact Factor: 1.392

## Metrics

• PDF downloads (101)
• HTML views (0)
• Cited by (16)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]