-
Previous Article
Emergent dynamics of an orientation flocking model for multi-agent system
- DCDS Home
- This Issue
-
Next Article
Sectional-hyperbolic Lyapunov stable sets
A class of anisotropic expanding curvature flows
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $ R^{n+1} $ with speed $ u^\alpha\sigma_k^\beta $ firstly, where $ u $ is support function of the hypersurface, $ \alpha, \beta \in R^1 $, and $ \beta>0 $, $ \sigma_k $ is the $ k $-th symmetric polynomial of the principal curvature radii of the hypersurface, $ k $ is an integer and $ 1\le k\le n $. For $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $ we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $, we prove that the flow with the speed $ fu^\alpha\sigma_k^\beta $ exists for all time and converges smoothly after normalisation to a soliton which is a solution of $ fu^{\alpha-1}\sigma_k^{\beta} = c $ provided that $ f $ is a smooth positive function on $ S^n $ and satisfies that $ (\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}}) $ is positive definite. When $ \beta = 1 $, our argument provides a proof to the well-known $ L_p $ Christoffel-Minkowski problem for the case $ p\ge k+1 $ where $ p = 2-\alpha $, which is identify with Ivaki's recent result. Especially, we obtain the same result for $ k = n $ without any constraint on smooth positive function $ f $. Finally, we also give a counterexample for the two anisotropic expanding flows when $ \alpha>1-k\beta $.
References:
[1] |
R. Alessandroni and C. Sinestrari,
Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 541-571.
|
[2] |
B. Andrews,
Entropy estimates for evolving hypersurfaces, Communications in Analysis and Geometry, 2 (1994), 267-275.
|
[3] |
B. Andrews,
Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.
doi: 10.1007/s002220050344. |
[4] |
B. Andrews,
Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171.
doi: 10.1007/BF01191340. |
[5] |
B. Andrews,
Monotone quantities and unique limits for evolving convex hypersurfaces, International Mathematics Research Notices, 20 (1997), 1001-1031.
doi: 10.1155/S1073792897000640. |
[6] |
B. Andrews and J. McCoy,
Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc., 364 (2012), 3427-3447.
doi: 10.1090/S0002-9947-2012-05375-X. |
[7] |
B. Andrews, J. McCoy and Y. Zheng,
Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations, 47 (2013), 611-665.
doi: 10.1007/s00526-012-0530-3. |
[8] |
S. Brendle, K. Choi and P. Daskalopoulos,
Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math., 219 (2017), 1-16.
doi: 10.4310/ACTA.2017.v219.n1.a1. |
[9] |
K. Choi and P. Daskalopoulos, Uniqueness of closed self-similar solutions to the Gauss curvature flow, arXiv: 1609.05487. |
[10] |
K.-S. Chou and X.-J. Wang,
A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 733-751.
doi: 10.1016/S0294-1449(00)00053-6. |
[11] |
K.-S. Chou and X.-J. Wang,
The $L_p$ Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. in Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[12] |
B. Chow,
Deforming convex hypersurfaces by the $n$-th root of the Gaussian curvature, J. Differential Geom., 22 (1985), 117-138.
doi: 10.4310/jdg/1214439724. |
[13] |
B. Chow,
Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math., 87 (1987), 63-82.
doi: 10.1007/BF01389153. |
[14] |
B. Chow and D.-H. Tsai,
Expansion of convex hypersurfaces by nonhomogeneous functions of curvature, Asian J. Math., 1 (1997), 769-784.
doi: 10.4310/AJM.1997.v1.n4.a7. |
[15] |
W. J. Firey,
Shapes of worn stones, Mathematika, 21 (1974), 1-11.
doi: 10.1112/S0025579300005714. |
[16] |
C. Gerhardt,
Non-scale-invariant inverse curvature flows in Euclidean space, Car. Var. Partial Differential Equations, 49 (2014), 471-489.
doi: 10.1007/s00526-012-0589-x. |
[17] |
P. F. Guan and C. S. Lin, On Equation $\det(u_ij+u\delta_ij) = u^{p}f $ on $S^n$, Preprint No 2000-7, NCTS in Tsing-Hua University, 2000. |
[18] |
P. F. Guan and X.-N. Ma,
Christoffel-Minkowski problem I: Convexity of solutions of a hessian equation, Invent. Math., 151 (2003), 553-577.
doi: 10.1007/s00222-002-0259-2. |
[19] |
P. F. Guan and L. Ni,
Entropy and a convergence theorem for Gauss curvature flow in high dimensions, J. Eur. Math. Soc., 19 (2017), 3735-3761.
doi: 10.4171/JEMS/752. |
[20] |
P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: The case $1 < p < k+1$, Cal. Var. Partial Differential Equations, 57 (2018), Art. 69, 23 pp.
doi: 10.1007/s00526-018-1341-y. |
[21] |
C. Q. Hu, X.-N. Ma and C. L. Shen,
On the Christoffel-Minkowski problem of Firey's $p$-sum, Cal. Var. Partial Differential Equations, 21 (2004), 137-155.
doi: 10.1007/s00526-003-0250-9. |
[22] |
Y. Huang, E. Lutwak, D. Yang and G. Y. 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. |
[23] |
G. Huisken,
Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
doi: 10.4310/jdg/1214438998. |
[24] |
M. N. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations, 58 (2019), Art. 1, 18 pp.
doi: 10.1007/s00526-018-1462-3. |
[25] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-010-9557-0. |
[26] |
Q.-R. Li, W. M. Sheng and X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., (2019).
doi: 10.4171/JEMS/936. |
[27] |
Q.-R. Li, W. M.Sheng and X.-J. Wang,
Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., 3 (2019), 1-27.
|
[28] |
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511526282.![]() ![]() ![]() |
[29] |
J. I. E. Urbas,
An expansion of convex hypersurfaces, J. Differential Geom., 33 (1991), 91-125.
doi: 10.4310/jdg/1214446031. |
[30] |
X.-J. Wang,
Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature, Trans. Amer. Math. Soc., 348 (1996), 4501-4524.
doi: 10.1090/S0002-9947-96-01650-9. |
[31] |
C. Xia,
Inverse anisotropic curvature flow from convex hypersurfaces, J. Geom. Anal., 27 (2017), 2131-2154.
doi: 10.1007/s12220-016-9755-2. |
show all references
References:
[1] |
R. Alessandroni and C. Sinestrari,
Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 541-571.
|
[2] |
B. Andrews,
Entropy estimates for evolving hypersurfaces, Communications in Analysis and Geometry, 2 (1994), 267-275.
|
[3] |
B. Andrews,
Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.
doi: 10.1007/s002220050344. |
[4] |
B. Andrews,
Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171.
doi: 10.1007/BF01191340. |
[5] |
B. Andrews,
Monotone quantities and unique limits for evolving convex hypersurfaces, International Mathematics Research Notices, 20 (1997), 1001-1031.
doi: 10.1155/S1073792897000640. |
[6] |
B. Andrews and J. McCoy,
Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc., 364 (2012), 3427-3447.
doi: 10.1090/S0002-9947-2012-05375-X. |
[7] |
B. Andrews, J. McCoy and Y. Zheng,
Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations, 47 (2013), 611-665.
doi: 10.1007/s00526-012-0530-3. |
[8] |
S. Brendle, K. Choi and P. Daskalopoulos,
Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math., 219 (2017), 1-16.
doi: 10.4310/ACTA.2017.v219.n1.a1. |
[9] |
K. Choi and P. Daskalopoulos, Uniqueness of closed self-similar solutions to the Gauss curvature flow, arXiv: 1609.05487. |
[10] |
K.-S. Chou and X.-J. Wang,
A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 733-751.
doi: 10.1016/S0294-1449(00)00053-6. |
[11] |
K.-S. Chou and X.-J. Wang,
The $L_p$ Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. in Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
[12] |
B. Chow,
Deforming convex hypersurfaces by the $n$-th root of the Gaussian curvature, J. Differential Geom., 22 (1985), 117-138.
doi: 10.4310/jdg/1214439724. |
[13] |
B. Chow,
Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math., 87 (1987), 63-82.
doi: 10.1007/BF01389153. |
[14] |
B. Chow and D.-H. Tsai,
Expansion of convex hypersurfaces by nonhomogeneous functions of curvature, Asian J. Math., 1 (1997), 769-784.
doi: 10.4310/AJM.1997.v1.n4.a7. |
[15] |
W. J. Firey,
Shapes of worn stones, Mathematika, 21 (1974), 1-11.
doi: 10.1112/S0025579300005714. |
[16] |
C. Gerhardt,
Non-scale-invariant inverse curvature flows in Euclidean space, Car. Var. Partial Differential Equations, 49 (2014), 471-489.
doi: 10.1007/s00526-012-0589-x. |
[17] |
P. F. Guan and C. S. Lin, On Equation $\det(u_ij+u\delta_ij) = u^{p}f $ on $S^n$, Preprint No 2000-7, NCTS in Tsing-Hua University, 2000. |
[18] |
P. F. Guan and X.-N. Ma,
Christoffel-Minkowski problem I: Convexity of solutions of a hessian equation, Invent. Math., 151 (2003), 553-577.
doi: 10.1007/s00222-002-0259-2. |
[19] |
P. F. Guan and L. Ni,
Entropy and a convergence theorem for Gauss curvature flow in high dimensions, J. Eur. Math. Soc., 19 (2017), 3735-3761.
doi: 10.4171/JEMS/752. |
[20] |
P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: The case $1 < p < k+1$, Cal. Var. Partial Differential Equations, 57 (2018), Art. 69, 23 pp.
doi: 10.1007/s00526-018-1341-y. |
[21] |
C. Q. Hu, X.-N. Ma and C. L. Shen,
On the Christoffel-Minkowski problem of Firey's $p$-sum, Cal. Var. Partial Differential Equations, 21 (2004), 137-155.
doi: 10.1007/s00526-003-0250-9. |
[22] |
Y. Huang, E. Lutwak, D. Yang and G. Y. 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. |
[23] |
G. Huisken,
Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
doi: 10.4310/jdg/1214438998. |
[24] |
M. N. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations, 58 (2019), Art. 1, 18 pp.
doi: 10.1007/s00526-018-1462-3. |
[25] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-010-9557-0. |
[26] |
Q.-R. Li, W. M. Sheng and X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., (2019).
doi: 10.4171/JEMS/936. |
[27] |
Q.-R. Li, W. M.Sheng and X.-J. Wang,
Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., 3 (2019), 1-27.
|
[28] |
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511526282.![]() ![]() ![]() |
[29] |
J. I. E. Urbas,
An expansion of convex hypersurfaces, J. Differential Geom., 33 (1991), 91-125.
doi: 10.4310/jdg/1214446031. |
[30] |
X.-J. Wang,
Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature, Trans. Amer. Math. Soc., 348 (1996), 4501-4524.
doi: 10.1090/S0002-9947-96-01650-9. |
[31] |
C. Xia,
Inverse anisotropic curvature flow from convex hypersurfaces, J. Geom. Anal., 27 (2017), 2131-2154.
doi: 10.1007/s12220-016-9755-2. |
[1] |
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 |
[2] |
Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025 |
[3] |
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 and Management Optimization, 2021 doi: 10.3934/jimo.2021211 |
[4] |
Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 |
[5] |
Anis Dhifaoui. $ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1403-1420. doi: 10.3934/dcdss.2022086 |
[6] |
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 and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 |
[7] |
Andrzej Świȩch. Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156 |
[8] |
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 |
[9] |
Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129 |
[10] |
Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036 |
[11] |
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 and Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021 |
[12] |
Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022030 |
[13] |
Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 |
[14] |
Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191 |
[15] |
Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210 |
[16] |
Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121 |
[17] |
Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075 |
[18] |
Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 |
[19] |
Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253 |
[20] |
Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $ L^p $-exact controllability of partial differential equations with nonlocal terms. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021053 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]