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On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains
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 |
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. |
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