\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

  • * Corresponding author: Peihe Wang

    * Corresponding author: Peihe Wang

The first author is supported by NSFC grant No. 11721101. The second author is supported by Shandong Provincial Natural Science Foundation ZR2020MA018

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we establish global $ C^2 $ a priori estimates for solutions to the uniformly parabolic equations with Neumann boundary condition on the smooth bounded domain in $ \mathbb R^n $ by a blow-up argument. As a corollary, we obtain that the solutions converge to ones which move by translation. This generalizes the viscosity results derived before by Da Lio.

    Mathematics Subject Classification: Primary: 35K55; Secondary: 35B40, 35K20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.
    [2] G. Barles and F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 521-541.  doi: 10.1016/j.anihpc.2004.09.001.
    [3] F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, J. Math. Anal. Appl., 339 (2008), 384-398.  doi: 10.1016/j.jmaa.2007.06.052.
    [4] Z. GaoX. MaP. Wang and L. Weng., Nonparametric mean curvature flow with nearly vertical contact angle condition, J. Math. Study, 54 (2021), 28-55.  doi: 10.4208/jms.v54n1.21.02.
    [5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
    [6] R. Huang and Y. Ye, A convergence result on the second boundary value problem for parabolic equations, Pacific J. Math., 310 (2021), 159-179.  doi: 10.2140/pjm.2021.310.159.
    [7] C. S. Kahane, A gradient estimate for solutions of the heat equation. II, Czechoslovak Math. J., 51 (2001), 39–44.. doi: 10.1023/A:1013745503001.
    [8] J. Kitagawa, A parabolic flow toward solutions of the optimal transportation problem on domains with boundary, J. Reine Angew. Math., 672 (2012), 127-160.  doi: 10.1515/crelle.2012.001.
    [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
    [10] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.
    [11] X.-N. MaP.-H. Wang and W. Wei, Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains, J. Funct. Anal., 274 (2018), 252-277.  doi: 10.1016/j.jfa.2017.10.002.
    [12] O. C. Schnürer, Translating solutions to the second boundary value problem for curvature flows, Manuscripta Math., 108 (2002), 319-347.  doi: 10.1007/s002290200265.
    [13] 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.
  • 加载中
SHARE

Article Metrics

HTML views(207) PDF downloads(253) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return