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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 |
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 $.
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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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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