American Institute of Mathematical Sciences

Advanced
doi: 10.3934/cpaa.2021198
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Boya Li 1, , Hongjie Ju 2, and  Yannan Liu 1,,

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 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 $.

Keywords: $ L_{p} $-Christoffel-Minkowski problem, existence of solutions, anisotropic curvature flow.
Mathematics Subject Classification: Primary: 35J96, 35J75; Secondary: 53A15, 53A07.
Citation: Boya Li, Hongjie Ju, Yannan Liu. A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem. Communications on Pure & Applied Analysis, 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.  Google Scholar

[2]

P. Bryan, M. 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.  Google Scholar

[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.  Google Scholar

[4]

C. Q. Chen, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. Hu, X. 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.  Google Scholar

[8]

Y. Huang, E. Lutwak, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[16]

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

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.  Google Scholar

[2]

P. Bryan, M. 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.  Google Scholar

[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.  Google Scholar

[4]

C. Q. Chen, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. Hu, X. 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.  Google Scholar

[8]

Y. Huang, E. Lutwak, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[16]

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

[1]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021
[2]

Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086
[3]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[4]

Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085
[5]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028
[6]

Xuerui Gao, Yanqin Bai, Shu-Cherng Fang, Jian Luo, Qian Li. A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021211
[7]

Burak Ordin, Adil Bagirov, Ehsan Mohebi. An incremental nonsmooth optimization algorithm for clustering using $ L_1 $ and $ L_\infty $ norms. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2757-2779. doi: 10.3934/jimo.2019079
[8]

Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367
[9]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[10]

Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254
[11]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051
[12]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[13]

K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013
[14]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058
[15]

Andrzej Świȩch. Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156
[16]

Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191
[17]

Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210
[18]

Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121
[19]

Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075
[20]

Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (78)
  • HTML views (66)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy