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

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 ∈ L_q/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.