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

Doyoon Kim; Seungjin Ryu

doi:10.3934/cpaa.2020024

Article Contents

2020, Volume 19, Issue 1: 493-510. Doi: 10.3934/cpaa.2020024

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

Doyoon Kim^1, and
Seungjin Ryu^2,

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

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)

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Abstract

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.

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Mathematics Subject Classification: 35B50, 35K20, 35J25.

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