TY - JOUR
T1 - The weak maximum principle for second-order elliptic and parabolic conormal derivative problems
AU - Kim, Doyoon
AU - Ryu, Seungjin
N1 - Funding Information:
2000 Mathematics Subject Classification. 35B50, 35K20, 35J25. Key words and phrases. Weak maximum principle, conormal derivative boundary condition, John domain. 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).
Publisher Copyright:
© 2020 American Institute of Mathematical Sciences. All rights reserved.
PY - 2020
Y1 - 2020
N2 - 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 Ln spaces (ai, bi ∈ Lq, c ∈ Lq/2, q = n if n ≥ 3 and q > 2 if n = 2). For the parabolic case, the lower-order coefficients ai, bi, and c belong to Lq,r spaces (ai, bi, |c|1/2 ∈ Lq,r with n/q + 2/r ≤ 1), q ∈ (n, ∞], r ∈ [2, ∞], n ≥ 2. We also consider coefficients in Ln,∞ with a smallness condition for parabolic equations.
AB - 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 Ln spaces (ai, bi ∈ Lq, c ∈ Lq/2, q = n if n ≥ 3 and q > 2 if n = 2). For the parabolic case, the lower-order coefficients ai, bi, and c belong to Lq,r spaces (ai, bi, |c|1/2 ∈ Lq,r with n/q + 2/r ≤ 1), q ∈ (n, ∞], r ∈ [2, ∞], n ≥ 2. We also consider coefficients in Ln,∞ with a smallness condition for parabolic equations.
KW - Conormal derivative boundary condition
KW - John domain
KW - Weak maximum principle
UR - http://www.scopus.com/inward/record.url?scp=85070772132&partnerID=8YFLogxK
U2 - 10.3934/cpaa.2020024
DO - 10.3934/cpaa.2020024
M3 - Article
AN - SCOPUS:85070772132
SN - 1534-0392
VL - 19
SP - 493
EP - 510
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 1
ER -