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Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation
Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains
1. | School of Mathematical Sciences, Nankai University, Tianjin 300071, China |
2. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
$ \Omega $ |
$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)^{s}u& = &f(u,v), & \mbox{in}\ \ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v& = &g(u,v),& \mbox{in}\ \ \Omega\,, \\[0.05cm] u,\,v&\equiv&0, & \mbox{on}\ \ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $ |
$ (u,\,v) $ |
References:
[1] |
G. Alberti and G. Bellettini,
A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560.
doi: 10.1007/s002080050159. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[3] |
H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42. |
[4] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Inequalities for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.
doi: 10.1215/S0012-7094-96-08117-X. |
[5] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.
doi: 10.1002/(SICI)1097-0312(199711)50:113.0.CO;2-6. |
[6] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.
|
[7] |
J. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[10] |
W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019). |
[11] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[12] |
W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. |
[13] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[14] |
Z. Chen, C. Lin and W. Zou,
Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.
doi: 10.1016/j.jfa.2013.08.021. |
[15] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[16] |
E. N. Dancer,
Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.
doi: 10.1007/s00208-008-0226-3. |
[17] |
S. Dipierro, N. Soave and E. Valdinoci,
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.
doi: 10.1007/s00208-016-1487-x. |
[18] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
[19] |
C. Li and Z. Wu,
Radial symmetry for systems of fractional Laplacian, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1567-1582.
doi: 10.1016/S0252-9602(18)30832-4. |
[20] |
L. Ma and Z. Zhang,
Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262.
doi: 10.1016/j.na.2018.12.015. |
[21] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
show all references
References:
[1] |
G. Alberti and G. Bellettini,
A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560.
doi: 10.1007/s002080050159. |
[2] |
D. Applebaum,
Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[3] |
H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42. |
[4] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Inequalities for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.
doi: 10.1215/S0012-7094-96-08117-X. |
[5] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.
doi: 10.1002/(SICI)1097-0312(199711)50:113.0.CO;2-6. |
[6] |
H. Berestycki, L. Caffarelli and L. Nirenberg,
Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.
|
[7] |
J. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[10] |
W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019). |
[11] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[12] |
W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. |
[13] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[14] |
Z. Chen, C. Lin and W. Zou,
Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.
doi: 10.1016/j.jfa.2013.08.021. |
[15] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[16] |
E. N. Dancer,
Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.
doi: 10.1007/s00208-008-0226-3. |
[17] |
S. Dipierro, N. Soave and E. Valdinoci,
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.
doi: 10.1007/s00208-016-1487-x. |
[18] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
[19] |
C. Li and Z. Wu,
Radial symmetry for systems of fractional Laplacian, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1567-1582.
doi: 10.1016/S0252-9602(18)30832-4. |
[20] |
L. Ma and Z. Zhang,
Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262.
doi: 10.1016/j.na.2018.12.015. |
[21] |
A. Quaas and A. Xia,
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659.
doi: 10.1007/s00526-014-0727-8. |
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