Before revision: We prove weak and strong maximum principles, including a Hopf lemma, for classical solutions to equations defined by linear, second-order, partial differential operators with non-negative characteristic form (degenerate-elliptic operators), in the presence of a second-order boundarycondition of Ventcel type along the degeneracy locus of the principle symbol of the operator on the domain boundary.
After revision: We prove weak and strong maximum principles, including a Hopf lemma, for classical solutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the principle symbol of the operator on the domain boundary.