Maximum principles for degenerate elliptic-parabolic equations with Venttsel’s boundary condition

Abstract

In this paper, we first establish interior and boundary maximum principles for degenerate elliptic-parabolic equations; we state both principles in one single theorem in terms of the propagation set (cf. Theorem 1). We next generalize the boundary condition to Venttsel’s one and obtain the similar result (cf. Theorem 2). Venttsel”s boundary condition contains Dirichlet, Neumann, oblique derivative and mixed boundary conditions as special cases and, from a probabilistic point of view (cf. Venttsel’ [9]), it is the most general admissible boundary condition. We give several examples in the last section.