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Zbl 0881.35049
Shi, Peter
Equivalence of variational inequalities with Wiener-Hopf equations. (English)
[J] Proc. Am. Math. Soc. 111, No.2, 339-346 (1991). ISSN 0002-9939; ISSN 1088-6826/e

The author compares a variational inequality $(Au,v-u)\ge(f,v-u)$ 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 $u=P(AP+Q)^{-1}f$.
[I.Bock (Bratislava)]

MSC 2000:: *35J85 Unilateral problems; variational inequalities (elliptic type)
35A15 Variational methods (PDE)
35K85 Unilateral problems; variational inequalities (parabolic type)

Keywords: convex cone; generalized Wiener-Hopf equation; projection operator; convergence of iteration scheme; parabolic variational inequalities with unilateral constraints

Cited in: Zbl 0912.49007

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