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

Doyoon Kim, Seungjin Ryu

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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 languageEnglish
Pages (from-to)493-510
Number of pages18
JournalCommunications on Pure and Applied Analysis
Volume19
Issue number1
DOIs
Publication statusPublished - 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

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