# American Institute of Mathematical Sciences

October  2021, 41(10): 4567-4592. doi: 10.3934/dcds.2021049

## On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

 1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

* Corresponding author: Xinghong Pan

Received  October 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

Fund Project: H. Dong was partially supported by the Simons Foundation, grant # 709545. X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. BK20180414) and National Natural Science Foundation of China (No. 11801268)

We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

Citation: Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4567-4592. doi: 10.3934/dcds.2021049
##### References:
 [1] P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.  doi: 10.1007/BF01759640.  Google Scholar [2] V. I. Bogachev and S. V. Shaposhnikov, Integrability and continuity of solutions to double divergence form equations, Ann. Mat. Pura Appl., 196 (2017), 1609-1635.  doi: 10.1007/s10231-016-0631-2.  Google Scholar [3] M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.  doi: 10.4064/cm-60-61-2-601-628.  Google Scholar [4] H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205 (2012), 119-149.  doi: 10.1007/s00205-012-0501-z.  Google Scholar [5] H. Dong and Z. Li, Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients, Trans. Amer. Math. Soc., 373 (2020), 4975-4997.  doi: 10.1090/tran/8042.  Google Scholar [6] H. Dong, J. Lee and S. Kim, On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients, Indiana U. Math. J., 69 (2020), 1815-1853.  doi: 10.1512/iumj.2020.69.8028.  Google Scholar [7] H. Dong, L. Escauriaza and S. Kim, On $C^1$, $C^2$, and weak type-(1, 1) estimates for linear elliptic operators: Part Ⅱ, Math. Ann., 370 (2018), 447-489.  doi: 10.1007/s00208-017-1603-6.  Google Scholar [8] H. Dong, L. Escauriaza and S. Kim, On $C^{1/2, 1}, C^{1, 2}$, and $C^{0, 0}$ estimates for linear parabolic operators, Preprint, arXiv: 1912.08762. Google Scholar [9] H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3.  Google Scholar [10] H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations, 40 (2011), 357-389.  doi: 10.1007/s00526-010-0344-0.  Google Scholar [11] H. Dong and S. Kim, On $C^1$, $C^2$, and weak type-$(1, 1)$ estimates for linear elliptic operators, Comm. Partial Differential Equations, 42 (2017), 417-435.  doi: 10.1080/03605302.2017.1278773.  Google Scholar [12] H. Dong and H. Zhang, Conormal problem of higher-order parabolic systems with time irregular coefficients, Trans. Amer. Math. Soc., 368 (2016), 7413-7460.  doi: 10.1090/tran/6605.  Google Scholar [13] L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.  Google Scholar [15] T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar [16] N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 250 (2007), 521-558.  doi: 10.1016/j.jfa.2007.04.003.  Google Scholar [17] Y. Y. Li, On the $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients, Chin. Ann. Math. Ser. B, 38 (2017), 489-496.  doi: 10.1007/s11401-017-1079-4.  Google Scholar [18] G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl., 148 (1987), 77-99.  doi: 10.1007/BF01774284.  Google Scholar [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathmetical Socierty: Providence, RI, 1968.  Google Scholar [20] M. I. Mati$\mathop {\rm{l}}\limits^ \vee$čuk and S. D. È$\mathop {\rm{l}}\limits^ \vee$del'man, On parabolic systems with coefficients satisfying Dini's condition, Dokl. Akad. Nauk SSSR, (Russian), 165 1965,482–485.  Google Scholar [21] V. Maz'ya and R. McOwen, Differentiability of solutions to second-order elliptic equations via dynamical systems, J. Differential Equations, 250 (2011), 1137-1168.  doi: 10.1016/j.jde.2010.06.023.  Google Scholar [22] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Ascillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

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##### References:
 [1] P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269.  doi: 10.1007/BF01759640.  Google Scholar [2] V. I. Bogachev and S. V. Shaposhnikov, Integrability and continuity of solutions to double divergence form equations, Ann. Mat. Pura Appl., 196 (2017), 1609-1635.  doi: 10.1007/s10231-016-0631-2.  Google Scholar [3] M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.  doi: 10.4064/cm-60-61-2-601-628.  Google Scholar [4] H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205 (2012), 119-149.  doi: 10.1007/s00205-012-0501-z.  Google Scholar [5] H. Dong and Z. Li, Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients, Trans. Amer. Math. Soc., 373 (2020), 4975-4997.  doi: 10.1090/tran/8042.  Google Scholar [6] H. Dong, J. Lee and S. Kim, On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients, Indiana U. Math. J., 69 (2020), 1815-1853.  doi: 10.1512/iumj.2020.69.8028.  Google Scholar [7] H. Dong, L. Escauriaza and S. Kim, On $C^1$, $C^2$, and weak type-(1, 1) estimates for linear elliptic operators: Part Ⅱ, Math. Ann., 370 (2018), 447-489.  doi: 10.1007/s00208-017-1603-6.  Google Scholar [8] H. Dong, L. Escauriaza and S. Kim, On $C^{1/2, 1}, C^{1, 2}$, and $C^{0, 0}$ estimates for linear parabolic operators, Preprint, arXiv: 1912.08762. Google Scholar [9] H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3.  Google Scholar [10] H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations, 40 (2011), 357-389.  doi: 10.1007/s00526-010-0344-0.  Google Scholar [11] H. Dong and S. Kim, On $C^1$, $C^2$, and weak type-$(1, 1)$ estimates for linear elliptic operators, Comm. Partial Differential Equations, 42 (2017), 417-435.  doi: 10.1080/03605302.2017.1278773.  Google Scholar [12] H. Dong and H. Zhang, Conormal problem of higher-order parabolic systems with time irregular coefficients, Trans. Amer. Math. Soc., 368 (2016), 7413-7460.  doi: 10.1090/tran/6605.  Google Scholar [13] L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.  Google Scholar [15] T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar [16] N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 250 (2007), 521-558.  doi: 10.1016/j.jfa.2007.04.003.  Google Scholar [17] Y. Y. Li, On the $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients, Chin. Ann. Math. Ser. B, 38 (2017), 489-496.  doi: 10.1007/s11401-017-1079-4.  Google Scholar [18] G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl., 148 (1987), 77-99.  doi: 10.1007/BF01774284.  Google Scholar [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathmetical Socierty: Providence, RI, 1968.  Google Scholar [20] M. I. Mati$\mathop {\rm{l}}\limits^ \vee$čuk and S. D. È$\mathop {\rm{l}}\limits^ \vee$del'man, On parabolic systems with coefficients satisfying Dini's condition, Dokl. Akad. Nauk SSSR, (Russian), 165 1965,482–485.  Google Scholar [21] V. Maz'ya and R. McOwen, Differentiability of solutions to second-order elliptic equations via dynamical systems, J. Differential Equations, 250 (2011), 1137-1168.  doi: 10.1016/j.jde.2010.06.023.  Google Scholar [22] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Ascillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, Princeton, NJ, 1993.  Google Scholar
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