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doi: 10.3934/dcds.2021152
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## Global $C^2$-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China 2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

* Corresponding author: Peihe Wang

Received  May 2020 Revised  January 2021 Early access October 2021

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

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.

Citation: Zhenghuan Gao, Peihe Wang. Global $C^2$-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021152
##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Z. Gao, X. Ma, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [10] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [11] X.-N. Ma, P.-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.  Google Scholar [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.  Google Scholar [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.  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. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Z. Gao, X. Ma, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [10] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [11] X.-N. Ma, P.-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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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