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 L-n spaces (a(i) , b(i) is an element of L-q , c is an element of 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 L-q,L-r spaces (a(i) , b(i) ,vertical bar c vertical bar(1/2) is an element of L-q,L-r with n/q + 2/r <= 1), q E (n, infinity], r is an element of [2, infinity], n >= 2. We also consider coefficients in L-n,L-infinity with a smallness condition for parabolic equations.