On the Generalized Stochastic Dirichlet Problem—Part II: Solvability, Stability and the Colombeau Case

Abstract

In this paper we consider the stochastic Dirichlet problem $L\lozenge u=h+\nabla f$ in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic in divergence form $L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u$. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space $W_0^{1,2}$ into the Kondratiev space (S)_− 1. In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes.