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January  2020, 19(1): 493-510. doi: 10.3934/cpaa.2020024

## The weak maximum principle for second-order elliptic and parabolic conormal derivative problems

 1 Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea 2 Department of Mathematics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul, 02504, Republic of Korea

Received  January 2019 Revised  May 2019 Published  July 2019

Fund Project: S. Ryu was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2017R1C1B1010966). D. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03934369).

We prove the weak maximum principle for second-order elliptic and parabolic equations in divergence form with the conormal derivative boundary conditions when the lower-order coefficients are unbounded and domains are beyond Lipschitz boundary regularity. In the elliptic case we consider John domains and lower-order coefficients in $L_n$ spaces ($a^i, b^i \in L_q$, $c \in L_{q/2}$, $q = n$ if $n \geq 3$ and $q > 2$ if $n = 2$). For the parabolic case, the lower-order coefficients $a^i$, $b^i$, and $c$ belong to $L_{q,r}$ spaces ($a^i,b^i, |c|^{1/2} \in L_{q,r}$ with $n/q+2/r \leq 1$), $q \in (n,\infty]$, $r \in [2,\infty]$, $n\ge 2$. We also consider coefficients in $L_{n,\infty}$ with a smallness condition for parabolic equations.

Citation: Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024
##### References:
 [1] H. Aikawa, Martin boundary and boundary Harnack principle for non-smooth domains, Selected papers on differential equations and analysis, 33–55, Amer. Math. Soc. Transl. Ser. 2,215, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/trans2/215/03. [2] B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, 52–68, Lecture Notes in Math., 1351, Springer, Berlin, 1988. doi: 10.1007/BFb0081242. [3] S. Bonafede and F. Nicolosi, A generalized maximum principle for boundary value problems for degenerate parabolic operators with discontinuous coefficients, Math. Bohem., 125 (2000), 39–54. [4] S. Buckley and P. Koskela, Sobolev-Poincaré implies John (English summary), Math. Res. Lett., 2 (1995), 577–593. doi: 10.4310/MRL.1995.v2.n5.a5. [5] S. Buckley and P. Koskela, Criteria for imbeddings of Sobolev-Poincaré type, Internat. Math. Res. Notices, 18 (1996), 881–901. doi: 10.1155/S1073792896000542. [6] E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [7] H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25–70. doi: 10.1007/s00205-009-0228-7. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [9] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557. [10] P. Hajłasz and P. Koskela, Sobolev embeddings, extensions and measure density condition (English summary), J. Funct. Anal., 254 (2008), 1217–1234. doi: 10.1016/j.jfa.2007.11.020. [11] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev met Poincaré (English summary), Mem. Amer. Math. Soc., 145 (2000). doi: 10.1090/memo/0688. [12] P. Hajłasz and O. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, J. Funct. Anal., 143 (1997), 221–246. doi: 10.1006/jfan.1996.2959. [13] D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172–202. doi: 10.1007/BF03041069. [14] D. A. Herron and P. Koskela, Uniform and Sobolev extension domains, Proc. Amer. Math. Soc., 114 (1992), 483–489. doi: 10.2307/2159672. [15] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71–88. doi: 10.1007/BF02392869. [16] H. Kim and M. Safonov, Boundary Harnack principle for second order elliptic equations with unbounded drift, Problems in Mathematical Analysis, No. 61. J. Math. Sci. (N.Y.) 179 (2011), 127–143. doi: 10.1007/s10958-011-0585-2. [17] H. Kim and M. Safonov, The boundary Harnack principle for second order elliptic equations in John and uniform domains, Proceedings of the St. Petersburg Mathematical Society, Vol. XV. Advances in Mathematical Analysis of Partial Differential Equations, 153–176, Amer. Math. Soc. Transl. Ser. 2,232, Amer. Math. Soc., Providence, RI, 2014. doi: 10.1090/trans2/232/09. [18] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968. [19] M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [20] M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679. [21] A. I. Nazarov and N. N. Ural'ceva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients (Russian); translated from Algebra i Analiz, 23 (2011), 136–168; St. Petersburg Math. J., 23 (2012), 93–115. doi: 10.1090/S1061-0022-2011-01188-4.

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##### References:
 [1] H. Aikawa, Martin boundary and boundary Harnack principle for non-smooth domains, Selected papers on differential equations and analysis, 33–55, Amer. Math. Soc. Transl. Ser. 2,215, Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/trans2/215/03. [2] B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, 52–68, Lecture Notes in Math., 1351, Springer, Berlin, 1988. doi: 10.1007/BFb0081242. [3] S. Bonafede and F. Nicolosi, A generalized maximum principle for boundary value problems for degenerate parabolic operators with discontinuous coefficients, Math. Bohem., 125 (2000), 39–54. [4] S. Buckley and P. Koskela, Sobolev-Poincaré implies John (English summary), Math. Res. Lett., 2 (1995), 577–593. doi: 10.4310/MRL.1995.v2.n5.a5. [5] S. Buckley and P. Koskela, Criteria for imbeddings of Sobolev-Poincaré type, Internat. Math. Res. Notices, 18 (1996), 881–901. doi: 10.1155/S1073792896000542. [6] E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [7] H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25–70. doi: 10.1007/s00205-009-0228-7. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [9] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557. [10] P. Hajłasz and P. Koskela, Sobolev embeddings, extensions and measure density condition (English summary), J. Funct. Anal., 254 (2008), 1217–1234. doi: 10.1016/j.jfa.2007.11.020. [11] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev met Poincaré (English summary), Mem. Amer. Math. Soc., 145 (2000). doi: 10.1090/memo/0688. [12] P. Hajłasz and O. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, J. Funct. Anal., 143 (1997), 221–246. doi: 10.1006/jfan.1996.2959. [13] D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172–202. doi: 10.1007/BF03041069. [14] D. A. Herron and P. Koskela, Uniform and Sobolev extension domains, Proc. Amer. Math. Soc., 114 (1992), 483–489. doi: 10.2307/2159672. [15] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71–88. doi: 10.1007/BF02392869. [16] H. Kim and M. Safonov, Boundary Harnack principle for second order elliptic equations with unbounded drift, Problems in Mathematical Analysis, No. 61. J. Math. Sci. (N.Y.) 179 (2011), 127–143. doi: 10.1007/s10958-011-0585-2. [17] H. Kim and M. Safonov, The boundary Harnack principle for second order elliptic equations in John and uniform domains, Proceedings of the St. Petersburg Mathematical Society, Vol. XV. Advances in Mathematical Analysis of Partial Differential Equations, 153–176, Amer. Math. Soc. Transl. Ser. 2,232, Amer. Math. Soc., Providence, RI, 2014. doi: 10.1090/trans2/232/09. [18] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968. [19] M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [20] M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679. [21] A. I. Nazarov and N. N. Ural'ceva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients (Russian); translated from Algebra i Analiz, 23 (2011), 136–168; St. Petersburg Math. J., 23 (2012), 93–115. doi: 10.1090/S1061-0022-2011-01188-4.
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