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. We apply these maximum principles to obtain uniqueness and a priori maximum principle estimates for classical solutions to boundary value and obstacle problems defined by these degenerate-elliptic operators , again in the presence of a second-order boundary condition, for Dirichlet or Neumann boundary conditions along the complement of the degeneracy locus.
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 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 smooth subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol vanishing locus. We obtain uniqueness and a priori maximum principle estimates for classical solutions to boundary value and obstacle problems defined by these degenerate-elliptic operators , again in the presence of a second-order boundary condition, for Dirichlet or Neumann boundary conditions along the complement of the degeneracy locus.