On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

We consider a very general second order nonlinear parabolic boundary value problem. Assuming the existence of an upper solution and a lower solutionψsatisfyingψ , we show that the problem has extremal periodic solutions in the order intervalK=[ψ, ]. Our proof is based on a general surjectivity result for the sum of two operators of monotone type and on truncation and penalization techniques. In addition we use a result of independent interest which we prove here and which says that the pseudomonotonicity property ofA(t, •) can be lifted to its Nemitsky operator. Finally when we impose stronger conditions on the data, we show that the extremal solutions can be obtained with a monotone iterative process.