ESTIMATE FOR EVOLUTIONARY SURFACES OF PRESCRIBED MEAN CURVATURE AND THE CONVERGENCE

Peihe Wang; Xinyu Gao

doi:10.11948/2018.1919

2018 Volume 8 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(6): 1919-1937

Next Article Previous Article

Peihe Wang, Xinyu Gao. ESTIMATE FOR EVOLUTIONARY SURFACES OF PRESCRIBED MEAN CURVATURE AND THE CONVERGENCE[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1919-1937. doi: 10.11948/2018.1919

Citation:

Peihe Wang, Xinyu Gao. ESTIMATE FOR EVOLUTIONARY SURFACES OF PRESCRIBED MEAN CURVATURE AND THE CONVERGENCE[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1919-1937. doi: 10.11948/2018.1919

ESTIMATE FOR EVOLUTIONARY SURFACES OF PRESCRIBED MEAN CURVATURE AND THE CONVERGENCE

School of Mathematical Sciences, Qufu Normal University, 57^#Jingxuan West Road, Qufu, 273165, China

Fund Project:

Abstract

Abstract

In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition f_τ ≥ -κ, which is the same as the one in the interior estimate by K. Ecker and generalizes the condition f_τ ≥ 0 studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.
- Mean curvature flow /
- gradient estimate /
- convergence
MSC: 35B45;35J92

FullText(HTML)

References

[1]	S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var., 1994, 2, 101-111. Google Scholar
[2]	K. A. Brakke, The motion of a surface by its mean curvature, in "Math. Notes," Princeton Univ. Press, Princeton, NJ, 1978. Google Scholar
[3]	P. Concus and R. Finn, On capillary free surface in a gravitational field, Acta Math., 1974, 132, 207-223. Google Scholar
[4]	K. Ecker, Estimates for evolutionary surfaces of prescribed mean curvature, Math. Z., 1982, 180, 179-192. Google Scholar
[5]	C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, J. Differ. Equat., 1980, 36, 139-172. Google Scholar
[6]	C. Gerhardt, Global regularity of the solutions to the capillary problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1976, 4(3), 151-176. Google Scholar
[7]	D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Google Scholar
[8]	B. Guan, Mean curvature motion of non-parametric hypersurfaces with contact angle condition, In Elliptic and Parabolic methods in Geometry, A K Peters, Wellesley(MA), 1996, 47-56. Google Scholar
[9]	B. Guan, Gradient estimates for solutions of nonparametric curvature evolution with prescribed contact angle condition, in Monge-Ampere Equation:Applications to Geometry and Optimization, Contemporary Mathematics, 1999, 226, 105-112. Google Scholar
[10]	G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 1984, 20, 237-266. Google Scholar
[11]	G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 1986, 84, 463-480. Google Scholar
[12]	G. Huisken, Non-parametric mean curvature evolution with boundary conditions, J. Differ. Equat., 1989, 77, 369-378. Google Scholar
[13]	H. Ishii, Introduction to Viscosity Solutions and the Large Time Behavior of Solutions:approximations, numerical analysis and applications, Lecture Notes in Math., 2074, 2013, 111-249. Google Scholar
[14]	N. J. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, Proc. Sympos. Pure Math., 1986, 45, Part 2:81-89. Google Scholar
[15]	N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Commu. Part. Differ. Equat., 1988, 13(1), 1-32. Google Scholar
[16]	O. A. Ladyzhenskaya and N. Ural'tseva, Local estimates for gradients of nonuniformly elliptic and parabolic equations, Comm.Pure Appl. Math., 1970, 23, 677-703. Google Scholar
[17]	G. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commu. Part. Differ. Equat., 1988, 13, 33-59. Google Scholar
[18]	G. Lieberman, Oblique boundary value problems for elliptic equations, World Scientific Publishing, 2013. Google Scholar
[19]	G. Lieberman, The first initial boundary value problem for quasilinear second order parabolic equations, Ann. Sci. Norm. Sup. Pisa Ser. IV, 1986, 8, 347-387. Google Scholar
[20]	A. Lichnewski and R. Temam, Surfaces minimales d'èvolutron:Le concept de pseudosolution, C.R. Acad. Sci. Paris, 1977, 284, 853-856. Google Scholar
[21]	P. L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Mathematical Journal, 1985, 52, 793-820. Google Scholar
[22]	X. N. Ma and J. J. Xu, Gradient estimates of mean curvature equations with Neumann boundary condition, Adv.Math., 2016, 290, 1010-1039. Google Scholar
[23]	X. N. Ma, P. H. Wang and W. Wei, Mean Curvature Equation and Mean Curvature Flow with Non-zero Neumann Boundary Conditions on Strictly Convex domain, Journal of Functional Analysis, 2018, 274, 252-277. Google Scholar
[24]	L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 1976, 61, 19-34. Google Scholar
[25]	J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 1975, 28, 189-200. Google Scholar
[26]	W. Sheng, N. Trudinger and X. J. Wang, Prescribed Weingarten Curvature Equations., Recent Development in Geometry and Analysis, ALM 2012, 23, 359-386. Google Scholar
[27]	O.C. Schnürer and R. S. Hartmut, Translating solutions for gauss curvature flows with Neumann boundary conditions, Pacific Journal of Mathematics, 2004, 213(1), 89-109. Google Scholar
[28]	P. H. Wang, The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height, Pacific Journal of Mathematics, 2014, 267(2), 489-509. Google Scholar
[29]	P. H. Wang, X. Liu and Z. H. Liu, The convexity of the level sets of maximal strictly space-like hypersurfaces defined on 2-dimensional space forms, Nonlinear Analysis, 2018, 174, 79-103. Google Scholar
[30]	P. H. Wang, H. M. Qiu and Z. H. Liu, Some geometrical properties of minimal graph on space forms with nonpositive curvature, Houston J. Math., 2018, 44(2), 545-570. Google Scholar
[31]	P. H. Wang and X. J. Wang,The geometric properties of harmonic functions on 2-dimensional Riemannian manifolds, Nonlinear Analysis, 2014, 103, 2-8. Google Scholar
[32]	P. H. Wang and L. L. Zhao,Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Nonlinear Analysis, 2016, 130, 1-17. Google Scholar
[33]	P. H. Wang and D. K. Zhang, Convexity of level sets of minimal graph on space form with nonnegative curvature with nonnegative curvature, J. Differ. Equat., 2017, 262, 5534-5564. Google Scholar
[34]	P. H. Wang and J. Zhuang, Convexity of level lines of maximal spacelike hypersurfaces in Minkowski space, Iseral J. Math., TBD 2018, 1-24. DOI:10.1007/s11856-018-1695-z. Google Scholar
[35]	X. J. Wang, Interior gradient estimates for mean curvature equations, Math.Z., 1998, 228, 73-81. Google Scholar
[36]	J. J. Xu, Mean curvature flows of graphs with Neumann boundary condition, https://arxiv.org/abs/1606.06392. Google Scholar