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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