• Previous Article
    The effect of nonlocal term on the superlinear elliptic equations in $ \mathbb{R}^{N} $
  • CPAA Home
  • This Issue
  • Next Article
    Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity
November  2019, 18(6): 3243-3265. doi: 10.3934/cpaa.2019146

Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems

1. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

2. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

* Corresponding author

Received  December 2018 Revised  February 2019 Published  May 2019

Fund Project: Research of the third author was supported by NSFC. No.11601311 and the fund of Shanghai Normal University

In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is $ u_{\nu} = -\phi(x)(1+|Du|^2)^\frac{1-q}{2} $ for any parameter $ q>0 $. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range $ 0<q<1 $, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any $ q>0 $. And in elliptic case, we generalize the results in [32] to any $ q\ge 0 $ and to any bounded smooth domain.

Citation: Jun Wang, Wei Wei, Jinju Xu. Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3243-3265. doi: 10.3934/cpaa.2019146
References:
[1]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var., 2 (1994), 101-111. doi: 10.1007/BF01234317.

[2]

B. Andrews and J. Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J., 58 (2009), 351-380. doi: 10.1512/iumj.2009.58.3756.

[3]

G. BarlesH. Ishii and H. Mitake, On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions, Arch. Rational Mech. Anal., 204 (2012), 515-558. doi: 10.1007/s00205-011-0484-1.

[4]

G. Barles and H. Mitake, A PDE approach to large-time asymptotics for boundary value problems for nonconvex Hamilton-Jacobi equations, Comm. in Partial Differential Equations, 37 (2012), 136-168. doi: 10.1080/03605302.2011.553645.

[5]

K. A. Brakke, The Motion of A Surface by Its Mean Curvature, Ph.D. Thesis, Princeton University, 1975.

[6]

P. Concus and R. Finn, On capillary free surfaces in the absence of gravity, Acta Math., 132 (1974), 177-198. doi: 10.1007/BF02392113.

[7]

C. M. EllottY. Giga and S. Goto, Dynamic boundary conditions for Hamilton-Jacobi equations, SIAM J. Math. Anal., 34 (2003), 861-881. doi: 10.1137/S003614100139957X.

[8]

R.Finn, Equilibrium Capillary Surfaces, Fundamental Principle of mathematics, 284, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8584-4.

[9]

C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Sci. Norm. Sup. Piss Ser. (4), 3 (1976), 157–175.

[10]

E. Giusti, On the equation of surfaces of prescribed mean curvature: existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137. doi: 10.1007/BF01393250.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag Berlin, 2001.

[12]

B. Guan, Mean curvature motion of non-parametric hypersurfaces with contact angle condition, in Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), AK Peters, Wellesley, MA, (1996), 47–56.

[13]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.

[14]

G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84 (1986), 463-480. doi: 10.1007/BF01388742.

[15]

G. Huisken, Non-parametric mean curvature evolution with boundary conditions, J. Differ. Equations., 77 (1989), 369-378. doi: 10.1016/0022-0396(89)90149-6.

[16]

L. Hormander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62 (1976), 1-52. doi: 10.1007/BF00251855.

[17]

H. Ishii, A short introduction to viscosity solutions and the large time behavior of solutions: approximations, numerical analysis and applications, in Lecture Notes in Math., 2074, Springer, Heidelberg, 111–249, 2013. doi: 10.1007/978-3-642-36433-4_3.

[18]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. in Partial Differential Equations, 13 (1988), 1-31. doi: 10.1080/03605308808820536.

[19]

G. M. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. in Partial Differential Equations, 13 (1988), 33-59. doi: 10.1080/03605308808820537.

[20]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.

[21]

G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Transactions of the American Mathematical Society, 295 (1986), 509-546. doi: 10.2307/2000050.

[22]

P. L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Mathematical Journal, 52 (1985), 793-820. doi: 10.1215/S0012-7094-85-05242-1.

[23]

H. Mitake, The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton- Jacobi equations, NoDEA Nonlinear Differential Equations App., 15 (2008), 347-362. doi: 10.1007/s00030-008-7043-y.

[24]

H. Mitake, Asymptotic solutions of Hamilton-Jacobi equations with state constraints, Appl. Math. Optim., 58 (2008), 393-410. doi: 10.1007/s00245-008-9041-1.

[25]

X. N. MaP. H. Wang and W. Wei, Mean curvature equation and mean curvature type flow with non-zero Neumann boundary conditions on strictly convex domains, J. Func. Anal., 274 (2018), 252-277. doi: 10.1016/j.jfa.2017.10.002.

[26]

X. N. Ma and J. J. Xu, Gradient estimates of mean curvature equations with Neumann boundary condition, Advances in Mathematics, 290 (2016), 1010-1039. doi: 10.1016/j.aim.2015.10.031.

[27]

O. C. Schnürer and R. S. Hartmut, Translating solutions for Gauss curvature flows with Neumann boundary conditions, Pacific J. Math., 213 (2004), 89-109. doi: 10.2140/pjm.2004.213.89.

[28]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34. doi: 10.1007/BF00251860.

[29]

J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200. doi: 10.1002/cpa.3160280202.

[30]

N. N. Ural'tseva, The solvability of the capillary problem, (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp., 4 (1973), 54–64.

[31]

X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81. doi: 10.1007/PL00004604.

[32]

J. J. Xu, A new proof of gradient estimates for mean curvature equations with oblique boundary conditions, Commun. Pure Appl. Anal., 15 (2016), 1719-1742. doi: 10.3934/cpaa.2016010.

[33]

J. J. Xu, Mean curvature flow of graphs with Neumann boundary conditions, Manuscripta Mathematica, 158 (2019), 75-84. doi: 10.1007/s00229-018-1007-2.

show all references

References:
[1]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var., 2 (1994), 101-111. doi: 10.1007/BF01234317.

[2]

B. Andrews and J. Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J., 58 (2009), 351-380. doi: 10.1512/iumj.2009.58.3756.

[3]

G. BarlesH. Ishii and H. Mitake, On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions, Arch. Rational Mech. Anal., 204 (2012), 515-558. doi: 10.1007/s00205-011-0484-1.

[4]

G. Barles and H. Mitake, A PDE approach to large-time asymptotics for boundary value problems for nonconvex Hamilton-Jacobi equations, Comm. in Partial Differential Equations, 37 (2012), 136-168. doi: 10.1080/03605302.2011.553645.

[5]

K. A. Brakke, The Motion of A Surface by Its Mean Curvature, Ph.D. Thesis, Princeton University, 1975.

[6]

P. Concus and R. Finn, On capillary free surfaces in the absence of gravity, Acta Math., 132 (1974), 177-198. doi: 10.1007/BF02392113.

[7]

C. M. EllottY. Giga and S. Goto, Dynamic boundary conditions for Hamilton-Jacobi equations, SIAM J. Math. Anal., 34 (2003), 861-881. doi: 10.1137/S003614100139957X.

[8]

R.Finn, Equilibrium Capillary Surfaces, Fundamental Principle of mathematics, 284, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8584-4.

[9]

C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Sci. Norm. Sup. Piss Ser. (4), 3 (1976), 157–175.

[10]

E. Giusti, On the equation of surfaces of prescribed mean curvature: existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137. doi: 10.1007/BF01393250.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag Berlin, 2001.

[12]

B. Guan, Mean curvature motion of non-parametric hypersurfaces with contact angle condition, in Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), AK Peters, Wellesley, MA, (1996), 47–56.

[13]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.

[14]

G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84 (1986), 463-480. doi: 10.1007/BF01388742.

[15]

G. Huisken, Non-parametric mean curvature evolution with boundary conditions, J. Differ. Equations., 77 (1989), 369-378. doi: 10.1016/0022-0396(89)90149-6.

[16]

L. Hormander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62 (1976), 1-52. doi: 10.1007/BF00251855.

[17]

H. Ishii, A short introduction to viscosity solutions and the large time behavior of solutions: approximations, numerical analysis and applications, in Lecture Notes in Math., 2074, Springer, Heidelberg, 111–249, 2013. doi: 10.1007/978-3-642-36433-4_3.

[18]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. in Partial Differential Equations, 13 (1988), 1-31. doi: 10.1080/03605308808820536.

[19]

G. M. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. in Partial Differential Equations, 13 (1988), 33-59. doi: 10.1080/03605308808820537.

[20]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.

[21]

G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Transactions of the American Mathematical Society, 295 (1986), 509-546. doi: 10.2307/2000050.

[22]

P. L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Mathematical Journal, 52 (1985), 793-820. doi: 10.1215/S0012-7094-85-05242-1.

[23]

H. Mitake, The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton- Jacobi equations, NoDEA Nonlinear Differential Equations App., 15 (2008), 347-362. doi: 10.1007/s00030-008-7043-y.

[24]

H. Mitake, Asymptotic solutions of Hamilton-Jacobi equations with state constraints, Appl. Math. Optim., 58 (2008), 393-410. doi: 10.1007/s00245-008-9041-1.

[25]

X. N. MaP. H. Wang and W. Wei, Mean curvature equation and mean curvature type flow with non-zero Neumann boundary conditions on strictly convex domains, J. Func. Anal., 274 (2018), 252-277. doi: 10.1016/j.jfa.2017.10.002.

[26]

X. N. Ma and J. J. Xu, Gradient estimates of mean curvature equations with Neumann boundary condition, Advances in Mathematics, 290 (2016), 1010-1039. doi: 10.1016/j.aim.2015.10.031.

[27]

O. C. Schnürer and R. S. Hartmut, Translating solutions for Gauss curvature flows with Neumann boundary conditions, Pacific J. Math., 213 (2004), 89-109. doi: 10.2140/pjm.2004.213.89.

[28]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34. doi: 10.1007/BF00251860.

[29]

J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200. doi: 10.1002/cpa.3160280202.

[30]

N. N. Ural'tseva, The solvability of the capillary problem, (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp., 4 (1973), 54–64.

[31]

X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81. doi: 10.1007/PL00004604.

[32]

J. J. Xu, A new proof of gradient estimates for mean curvature equations with oblique boundary conditions, Commun. Pure Appl. Anal., 15 (2016), 1719-1742. doi: 10.3934/cpaa.2016010.

[33]

J. J. Xu, Mean curvature flow of graphs with Neumann boundary conditions, Manuscripta Mathematica, 158 (2019), 75-84. doi: 10.1007/s00229-018-1007-2.

[1]

Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010

[2]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[3]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[4]

Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139

[5]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69

[6]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237

[7]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441

[8]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[9]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[10]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

[11]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[12]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[13]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[14]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[15]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

[16]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216

[17]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[18]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[19]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[20]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (12)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]