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 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.
Original language | English |
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Pages (from-to) | 493-510 |
Number of pages | 18 |
Journal | Communications on Pure and Applied Analysis |
Volume | 19 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2020 |
Bibliographical note
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.
Keywords
- Conormal derivative boundary condition
- John domain
- Weak maximum principle
ASJC Scopus subject areas
- Analysis
- Applied Mathematics