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

Doyoon Kim, Seungjin Ryu

Research output: Contribution to journalArticlepeer-review

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 Jan 1

Keywords

  • Conormal derivative boundary condition
  • John domain
  • Weak maximum principle

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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