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