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A class of random variational inequalities and simple random unilateral boundary value problems: Existence, discretization, finite element approximation. (English) Zbl 0974.60048
At first the author proves the existence of a unique measurable solution for the following random variational inequality: Find for every elementary event ω an element y ^𝒦 depending on ω such that β(ω,y ^,z-y ^)λ(ω,z-y ^), z𝒦, where 𝒦 is a closed convex non-empty subset of a separable Hilbert space and for every ω, λ(ω,·) and β(ω,·,·) being a linear form, respectively, bilinear form. It is assumed that β is nonnegative. Then the author considers in detail a more specialized inequality, where the data decompose in deterministic data and given real-valued random variables. Existence and uniqueness results are given, various approximation procedures and its convergence are studied. The general theory is applied to a Helmholtz-like elliptic equation with Signorini boundary conditions.

MSC:
60H25Random operators and equations