## 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 (a^{i}, b^{i} ∈ Lq, c ∈ L_{q/}2, q = n if n ≥ 3 and q > 2 if n = 2). For the parabolic case, the lower-order coefficients a^{i}, b^{i}, and c belong to Lq,r spaces (a^{i}, b^{i}, |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 Jan 1 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics