Generalized Wentzell boundary conditions for second order operators with interior degeneracy

We consider operators in divergence form, A1u = (au')', and in nondivergence form, A2u = au", provided that the coefficient a vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators A1 and A2, equipped with general Wentzell boundary conditions, are nonpositive and self-adjoint on spaces of L2type