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Equivalence of variational inequalities with Wiener-Hopf equations. (English) Zbl 0881.35049
The author compares a variational inequality \((Au,v-u)\geq(f,v-u)\) for all \(v\in K\) and a generalized Wiener-Hopf equation \((AP+Q)v=f\), where \(A:D(A)\to H\) is an arbitrary operator, \(H\) is a Hilbert space, \(K\) its closed convex subset, \(P\) the projection operator from \(H\) into \(K\), \(Q=I-P\). The main results are as follows: The variational inequality has a solution \(u\) if and only if the Wiener-Hopf equation has a solution \(v\), \(v=u+f-Au\), \(u=Pv\). If a solution \(u\) is unique for each \(f\), then \(u=P(AP+Q)^{-1}f\).

MSC:
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35A15 Variational methods applied to PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
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