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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 & 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.  Google Scholar [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.  Google Scholar [3] G. Barles, H. 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.  Google Scholar [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.  Google Scholar [5] K. A. Brakke, The Motion of A Surface by Its Mean Curvature, Ph.D. Thesis, Princeton University, 1975.  Google Scholar [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.  Google Scholar [7] C. M. Ellott, Y. Giga and S. Goto, Dynamic boundary conditions for Hamilton-Jacobi equations, SIAM J. Math. Anal., 34 (2003), 861-881.  doi: 10.1137/S003614100139957X.  Google Scholar [8] R.Finn, Equilibrium Capillary Surfaces, Fundamental Principle of mathematics, 284, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8584-4.  Google Scholar [9] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Sci. Norm. Sup. Piss Ser. (4), 3 (1976), 157–175.  Google Scholar [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.  Google Scholar [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag Berlin, 2001.  Google Scholar [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.  Google Scholar [13] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.   Google Scholar [14] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84 (1986), 463-480.  doi: 10.1007/BF01388742.  Google Scholar [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.  Google Scholar [16] L. Hormander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62 (1976), 1-52.  doi: 10.1007/BF00251855.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] X. N. Ma, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] G. Barles, H. 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.  Google Scholar [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.  Google Scholar [5] K. A. Brakke, The Motion of A Surface by Its Mean Curvature, Ph.D. Thesis, Princeton University, 1975.  Google Scholar [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.  Google Scholar [7] C. M. Ellott, Y. Giga and S. Goto, Dynamic boundary conditions for Hamilton-Jacobi equations, SIAM J. Math. Anal., 34 (2003), 861-881.  doi: 10.1137/S003614100139957X.  Google Scholar [8] R.Finn, Equilibrium Capillary Surfaces, Fundamental Principle of mathematics, 284, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8584-4.  Google Scholar [9] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Sci. Norm. Sup. Piss Ser. (4), 3 (1976), 157–175.  Google Scholar [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.  Google Scholar [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag Berlin, 2001.  Google Scholar [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.  Google Scholar [13] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.   Google Scholar [14] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84 (1986), 463-480.  doi: 10.1007/BF01388742.  Google Scholar [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.  Google Scholar [16] L. Hormander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62 (1976), 1-52.  doi: 10.1007/BF00251855.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] X. N. Ma, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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