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February  2016, 36(2): 971-980. doi: 10.3934/dcds.2016.36.971

Topological degree method for the rotationally symmetric $L_p$-Minkowski problem

Jian Lu 1, and  Huaiyu Jian 2,

1. 

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  October 2014 Revised  February 2015 Published  August 2015

Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
Keywords: topological degree, existence of solutions., Monge-Ampère equation, Minkowski problem, centro-affine Gauss curvature.
Mathematics Subject Classification: Primary: 35J96, 35J75, 53A15; Secondary: 34C4.
Citation: Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
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.  Google Scholar

[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.  Google Scholar

[3]

B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371. doi: 10.1007/s005260050111.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.  Google Scholar

[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.  Google Scholar

[19]

E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16.  Google Scholar

[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.  Google Scholar

[21]

G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975.  Google Scholar

[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.  Google Scholar

[23]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.  Google Scholar

[24]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations, 7 (1998), 315-371. doi: 10.1007/s005260050111.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.  Google Scholar

[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.  Google Scholar

[19]

E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Diff. Geom., 47 (1997), 1-16.  Google Scholar

[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.  Google Scholar

[21]

G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, R.I., fourth edition, 1975.  Google Scholar

[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.  Google Scholar

[23]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.  Google Scholar

[24]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174.  Google Scholar

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