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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 and 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.

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