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March  2022, 21(3): 785-796. doi: 10.3934/cpaa.2021198

A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

1. 

School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China

2. 

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

*Corresponding author

Received  July 2021 Revised  October 2021 Published  March 2022 Early access  December 2021

Fund Project: This work was supported in part by the Natural Science Foundation of Beijing Municipality (No. 1212002) and the National Natural Science Foundation of China (Grant Nos. 12071017, 11871432, 11871102)

In this paper, a generalitzation of the $ L_{p} $-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for $ c = 1 $.

Citation: Boya Li, Hongjie Ju, Yannan Liu. A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem. Communications on Pure and Applied Analysis, 2022, 21 (3) : 785-796. doi: 10.3934/cpaa.2021198
References:
[1]

K. J. Boroczky and F. Fodor, $L_p$ dual Minkowski prolblem for $p > 1$ and $q > 0$, J. Differ. Equ., 266 (2019), 7980-8033.  doi: 10.1016/j.jde.2018.12.020.

[2]

P. BryanM. N. Ivaki and J. Scheuer, A unified flow approach to smooth, even $L_p$-Minkowski problems, Anal. PDE, 12 (2019), 259-280.  doi: 10.2140/apde.2019.12.259.

[3]

P. Bryan, M. N. Ivaki and J. Scheuer, Orlicz-Minkowski flows, Calc. Var. Partial Differ. Equ., 60 (2021), 25pp. doi: 10.1007/s00526-020-01886-3.

[4]

C. Q. ChenY. Huang and Y. M. Zhao, Smooth solutions to the $L_{p}$ dual Minkowski problems, Math. Ann., 373 (2019), 953-976.  doi: 10.1007/s00208-018-1727-3.

[5]

P. F. Guan and X. N. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of Hessian equation, Invent. Math., 151 (2003), 553-571.  doi: 10.1007/s00222-002-0259-2.

[6]

P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: the case $1 < p < k+1$, Calc. Var. Partial Differ. Equ., 57 (2018), 23 pp. doi: 10.1007/s00526-018-1341-y.

[7]

C. HuX. N. Ma and C. Shen, On Christoffel-Minkowski problem of Firey's $p$-sum, Calc. Var. Partial Differ. Equ., 21 (2004), 137-155.  doi: 10.1007/s00526-003-0250-9.

[8]

Y. HuangE. LutwakD. Yang and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216 (2016), 325-388.  doi: 10.1007/s11511-016-0140-6.

[9]

Y. Huang and Y. M. Zhao, On the $L_p$ dual Minkowski problem, Adv. Math., 332 (2018), 57-84.  doi: 10.1016/j.aim.2018.05.002.

[10]

M. N. Ivaki, Deforming a hyper surface by principal radii of curvature and support function, Calc. Var. Partial Differ. Equ., 58 (2019), 2133-2165.  doi: 10.1007/s00526-018-1462-3.

[11]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 239 (1980), 161-175.  doi: 10.1070/IM1981v016n01ABEH001283.

[12]

Y. N. Liu and J. Lu, A flow method for the dual Orlicz-Minkowski problem, Trans. Amer. Math. Soc., 373 (2020), 5833-5853.  doi: 10.1090/tran/8130.

[13]

E. Lutwak, D. Yang and G. Y. Zhang, $L_{p}$ dual curvature measures, Adv. Math., 329 (2018), 85-132. doi: 10.1016/j.aim.2018.02.011.

[14] R. Schneider, Convex bodies, the Brunn-Minkowski theory, in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, expanded, 2014. 
[15]

W. M. Sheng and C. H. Yi, A class of anisotropic expanding curvature flow, Disc. Conti. Dynam. Systems-A, 40 (2020), 2017-2035.  doi: 10.3934/dcds.2020104.

[16]

J. Urbas, An expansion of convex hypersurfaces, J. Differ. Geom., 33 (1991), 91-125.  doi: 10.4310/jdg/1214446031.

show all references

References:
[1]

K. J. Boroczky and F. Fodor, $L_p$ dual Minkowski prolblem for $p > 1$ and $q > 0$, J. Differ. Equ., 266 (2019), 7980-8033.  doi: 10.1016/j.jde.2018.12.020.

[2]

P. BryanM. N. Ivaki and J. Scheuer, A unified flow approach to smooth, even $L_p$-Minkowski problems, Anal. PDE, 12 (2019), 259-280.  doi: 10.2140/apde.2019.12.259.

[3]

P. Bryan, M. N. Ivaki and J. Scheuer, Orlicz-Minkowski flows, Calc. Var. Partial Differ. Equ., 60 (2021), 25pp. doi: 10.1007/s00526-020-01886-3.

[4]

C. Q. ChenY. Huang and Y. M. Zhao, Smooth solutions to the $L_{p}$ dual Minkowski problems, Math. Ann., 373 (2019), 953-976.  doi: 10.1007/s00208-018-1727-3.

[5]

P. F. Guan and X. N. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of Hessian equation, Invent. Math., 151 (2003), 553-571.  doi: 10.1007/s00222-002-0259-2.

[6]

P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: the case $1 < p < k+1$, Calc. Var. Partial Differ. Equ., 57 (2018), 23 pp. doi: 10.1007/s00526-018-1341-y.

[7]

C. HuX. N. Ma and C. Shen, On Christoffel-Minkowski problem of Firey's $p$-sum, Calc. Var. Partial Differ. Equ., 21 (2004), 137-155.  doi: 10.1007/s00526-003-0250-9.

[8]

Y. HuangE. LutwakD. Yang and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216 (2016), 325-388.  doi: 10.1007/s11511-016-0140-6.

[9]

Y. Huang and Y. M. Zhao, On the $L_p$ dual Minkowski problem, Adv. Math., 332 (2018), 57-84.  doi: 10.1016/j.aim.2018.05.002.

[10]

M. N. Ivaki, Deforming a hyper surface by principal radii of curvature and support function, Calc. Var. Partial Differ. Equ., 58 (2019), 2133-2165.  doi: 10.1007/s00526-018-1462-3.

[11]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 239 (1980), 161-175.  doi: 10.1070/IM1981v016n01ABEH001283.

[12]

Y. N. Liu and J. Lu, A flow method for the dual Orlicz-Minkowski problem, Trans. Amer. Math. Soc., 373 (2020), 5833-5853.  doi: 10.1090/tran/8130.

[13]

E. Lutwak, D. Yang and G. Y. Zhang, $L_{p}$ dual curvature measures, Adv. Math., 329 (2018), 85-132. doi: 10.1016/j.aim.2018.02.011.

[14] R. Schneider, Convex bodies, the Brunn-Minkowski theory, in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, expanded, 2014. 
[15]

W. M. Sheng and C. H. Yi, A class of anisotropic expanding curvature flow, Disc. Conti. Dynam. Systems-A, 40 (2020), 2017-2035.  doi: 10.3934/dcds.2020104.

[16]

J. Urbas, An expansion of convex hypersurfaces, J. Differ. Geom., 33 (1991), 91-125.  doi: 10.4310/jdg/1214446031.

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