A Generalized Maximum Principle for Boundary Value Problems for Degenerate Parabolic Operators with Discontinuous Coefficients

In [14] M.G.Platone Garroni has extended the classical generalized maximum principle (see, for instance, [15]), when the coefficients of the operator are discontinuous, to subsolutions of elliptic linear second order equations with mixed type boundary unilateral conditions, that is, on a portion of the boundary ∂Ω of Ω, the values of the solution are assigned, while on the other part a unilateral condition on the solution and its conormal derivative is given. In the present paper we will establish a similar result (see Theorem 5.1) for degenerate parabolic equations, using a technique different from that of [14]. As a corollary, we obtain a comparison theorem (see Theorem 6.1). Our procedure, rather similar to that followed in [12] and in [13] allows us to obtain more general results. Other sufficient conditions for the boundedness of weak subsolutions of Cauchy-Dirichlet problem, in the non degenerate case, may be obtained from [6] and [17], while in the degenerate case some results are announced in [3] and in [4].