Topological degree method for the rotationally symmetric L_p-Minkowski problem

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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

Abstract
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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 and 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.
[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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