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Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment
Topological degree method for the rotationally symmetric $L_p$-Minkowski problem
1. | Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China |
2. | Department of Mathematical Sciences, Tsinghua University, Beijing 100084 |
References:
[1] |
J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337.
doi: 10.1007/s005260000075. |
[2] |
L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257.
doi: 10.1007/BF00375127. |
[3] |
B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371.
doi: 10.1007/s005260050111. |
[4] |
J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.
doi: 10.1090/S0894-0347-2012-00741-3. |
[5] |
E. Calabi, Complete affine hyperspheres. I, in Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, (1972), 19-38. |
[6] |
S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[7] |
W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89.
doi: 10.1016/j.aim.2004.11.007. |
[8] |
W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667.
doi: 10.1002/cpa.3160480606. |
[9] |
K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[10] |
K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420035704. |
[11] |
J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, Adv. Math., 230 (2012), 1209-1221.
doi: 10.1016/j.aim.2012.02.027. |
[12] |
M. Ji, On positive scalar curvature on $S^2$, Calc. Var. Partial Differential Equations, 19 (2004), 165-182.
doi: 10.1007/s00526-003-0214-0. |
[13] |
H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations, J. Diff. Geom., 93 (2013), 431-469. |
[14] |
M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.
doi: 10.1007/s00526-010-0375-6. |
[15] |
Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[16] |
J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differential Equations, 254 (2013), 983-1005.
doi: 10.1016/j.jde.2012.10.008. |
[17] |
E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150. |
[18] |
E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc., 356 (2004), 4359-4370.
doi: 10.1090/S0002-9947-03-03403-2. |
[19] |
E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16. |
[20] |
R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations, 4 (1996), 1-25.
doi: 10.1007/BF01322307. |
[21] |
G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975. |
[22] |
V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math., 180 (2003), 176-186.
doi: 10.1016/S0001-8708(02)00101-9. |
[23] |
G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931.
doi: 10.1016/j.aim.2014.06.004. |
[24] |
G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174. |
show all references
References:
[1] |
J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337.
doi: 10.1007/s005260000075. |
[2] |
L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257.
doi: 10.1007/BF00375127. |
[3] |
B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371.
doi: 10.1007/s005260050111. |
[4] |
J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.
doi: 10.1090/S0894-0347-2012-00741-3. |
[5] |
E. Calabi, Complete affine hyperspheres. I, in Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, (1972), 19-38. |
[6] |
S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[7] |
W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89.
doi: 10.1016/j.aim.2004.11.007. |
[8] |
W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667.
doi: 10.1002/cpa.3160480606. |
[9] |
K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[10] |
K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420035704. |
[11] |
J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, Adv. Math., 230 (2012), 1209-1221.
doi: 10.1016/j.aim.2012.02.027. |
[12] |
M. Ji, On positive scalar curvature on $S^2$, Calc. Var. Partial Differential Equations, 19 (2004), 165-182.
doi: 10.1007/s00526-003-0214-0. |
[13] |
H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations, J. Diff. Geom., 93 (2013), 431-469. |
[14] |
M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.
doi: 10.1007/s00526-010-0375-6. |
[15] |
Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[16] |
J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differential Equations, 254 (2013), 983-1005.
doi: 10.1016/j.jde.2012.10.008. |
[17] |
E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150. |
[18] |
E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc., 356 (2004), 4359-4370.
doi: 10.1090/S0002-9947-03-03403-2. |
[19] |
E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16. |
[20] |
R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations, 4 (1996), 1-25.
doi: 10.1007/BF01322307. |
[21] |
G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975. |
[22] |
V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem, Adv. Math., 180 (2003), 176-186.
doi: 10.1016/S0001-8708(02)00101-9. |
[23] |
G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931.
doi: 10.1016/j.aim.2014.06.004. |
[24] |
G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174. |
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