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Liouville theorems on the upper half space
1. | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
2. | School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China |
3. | Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA |
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $ a \in (0, 1) $ constants are the only $ C^1 $ up to the boundary positive solutions to $ div(x_n^a \nabla u) = 0 $ on the upper half space.
References:
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doi: 10.1007/978-1-4757-8137-3. |
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Short proofs of three theorems on harmonic functions, Proc. Amer. Math. Soc., 102 (1988), 906-908.
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L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
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T. Carleman,
Zur Theorie de Minimalflächen, Math. Z., 9 (1921), 154-160.
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S. Chen,
A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, 2014 (2012), 1205-1220.
doi: 10.1093/imrn/rns248. |
[6] |
J. Dou, Q. Guo and M. Zhu,
Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.
doi: 10.1016/j.aim.2017.03.007. |
[7] |
J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924. |
[8] |
J. Dou and M. Zhu,
Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 2015 (2013), 651-687.
doi: 10.1093/imrn/rnt213. |
[9] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
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M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 46pp.
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F. Hang, X. Wang and X. Yan,
Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.
doi: 10.1002/cpa.20193. |
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Y. Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
show all references
References:
[1] |
S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics, 137, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-8137-3. |
[2] |
H. P. Boas and R. P. Boas,
Short proofs of three theorems on harmonic functions, Proc. Amer. Math. Soc., 102 (1988), 906-908.
doi: 10.1090/S0002-9939-1988-0934865-6. |
[3] |
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. |
[4] |
T. Carleman,
Zur Theorie de Minimalflächen, Math. Z., 9 (1921), 154-160.
doi: 10.1007/BF01378342. |
[5] |
S. Chen,
A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, 2014 (2012), 1205-1220.
doi: 10.1093/imrn/rns248. |
[6] |
J. Dou, Q. Guo and M. Zhu,
Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.
doi: 10.1016/j.aim.2017.03.007. |
[7] |
J. Dou, L. Sun, L. Wang and M. Zhu, Divergent operator with degeneracy and related sharp inequalities, preprint, arXiv: 1910.13924. |
[8] |
J. Dou and M. Zhu,
Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 2015 (2013), 651-687.
doi: 10.1093/imrn/rnt213. |
[9] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[10] |
M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 46pp.
doi: 10.1016/j.jfa.2018.08.012. |
[11] |
F. Hang, X. Wang and X. Yan,
Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.
doi: 10.1002/cpa.20193. |
[12] |
Y. Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
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