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 \(u=P(AP+Q)^{-1}f\).


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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[1] Claudio Baiocchi, Fabio Gastaldi, and Franco Tomarelli, Some existence results on noncoercive variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 4, 617 – 659. · Zbl 0644.49004
[2] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. · Zbl 0331.35002
[3] Avner Friedman, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. · Zbl 0564.49002
[4] Vasile I. Istrăţescu, Fixed point theory, Mathematics and its Applications, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. An introduction; With a preface by Michiel Hazewinkel.
[5] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0457.35001
[6] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, Springer-Verlag, Berlin, 1972. · Zbl 0227.35001
[7] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493 – 519. · Zbl 0152.34601
[8] A. Pitonyak, P. Shi, and M. Shillor, Numerical solutions to obstacle problems by a new iteration scheme, preprint.
[9] Siegfried Prössdorf, Einige Klassen singulärer Gleichungen, Birkhäuser Verlag, Basel-Stuttgart, 1974 (German). Mathematische Reihe, Band 46. · Zbl 0302.45008
[10] José-Francisco Rodrigues, Obstacle problems in mathematical physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 114. · Zbl 0606.73017
[11] F.-O. Speck, General Wiener-Hopf factorization methods, Research Notes in Mathematics, vol. 119, Pitman (Advanced Publishing Program), Boston, MA, 1985. With a foreword by E. Meister. · Zbl 0588.35090
[12] D. V. Widder, The heat equation, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 67. · Zbl 0322.35041
[13] Eduardo H. Zarantonello , Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971. Mathematics Research Center, Publ. No. 27. · Zbl 0263.00001
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