zbMATH — the first resource for mathematics

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.