Variational inequalities

The variational formulations (also called weak formulations) of many non-linear boundary value problems result in variational inequalities rather than variational equations. Analogously to partial differential equations, variational inequalities can be of elliptic, parabolic, hyperbolic, etc. type.

Suppose that $V$ is a Banach space, $V'$ is its dual and $(.,.)$ is the duality pairing between $V$ and $V'$. Let $a(.,.):V\times V\to\mathbf{R}$ be a continuous, bilinear, coercive or semi-coercive form (cf. also Coerciveness inequality), let $l\in V'$, and let $K\subset V$ be a non-empty, convex, closed subset. Moreover, let $\Phi:V\to(-\infty,+\infty]$, $\Phi\not\equiv\infty$, be a convex, lower semi-continuous functional on $V$.

Then an elliptic variational inequality usually has one of the following two forms:

1) Find $u\in K$ such that

\begin{equation}a(u,v-u)\geq(l,v-u), \forall v\in K,\end{equation}

2) Find $u\in K$ such that

\begin{equation}a(u,v-u)+\Phi(v)-\Phi(u)\geq(l,v-u), \forall v\in V.\end{equation}

Eigenvalue problems related to variational inequalities have been formulated and studied, as well as optimal control problems of system governed by, i.e. having as state relations, variational inequalities. In mechanics, variational inequalities arise in a natural way because they are expressions of the principle of virtual work or power in inequality form. The mathematical study of variational inequalities began in the early 1960s with the work of G. Fichera, J.L. Lions and G. Stampacchia [a1], [a2]. J.J. Moreau [a3] and H. Brézis [a4] connected the theory of variational inequalities to convex analysis, especially to the notion of subdifferentiability, and to the theory of maximal monotone operators [a5]. For non-coercive variational inequalities see [a6], [a7]; for their numerical study, see [a8], [a9]. Eigenvalue problems for variational inequalities can be found in [a10], while applications can be found in [a11], [a12], where also their relation to convex non-smooth energy minimization problems is presented. For the optimal control problem of systems governed by variational inequalities see [a13].

References

[a1]	G. Fichera, "Problemi Elastostatici con Vincoli Unilaterali, il Problema di Signorini con Ambigue Condizioni al Contorno" Mem. Accad. Naz. Lincei, VIII , 7 (1964) pp. 91–140
[a2]	J.L. Lions, G. Stampacchia, "Variational inequalities" Commun. Pure Appl. Math. , XX (1967) pp. 493–519
[a3]	J.J. Moreau, "Fonctionnelles convexes. Sém. sur les Équations aux Dérivées Partielles" , Collége de France (1967)
[a4]	H. Brézis, "Problèmes unilatéraux" J. Math. Pures Appl. , 51 (1972) pp. 1–168
[a5]	H. Brézis, "Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert" , North-Holland &Amer. Elsevier (1973)
[a6]	C. Baiocchi, F. Gastaldi, F. Tomarelli, "Some existence results on noncoercive variational inequalities" Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV , 13 (1986) pp. 617–659
[a7]	D. Goeleven, "Noncoercive variational problems and related results" , Res. Notes Math. Sci. , 357 , Longman (1996)
[a8]	R. Glowinski, J.L. Lions, R. Trémoliéres, "Analyse numérique des inéquations variationnelles" , Dunod (1976)
[a9]	J. Haslinger, I. Hlavaček, J. Nečas, "Numerical methods for unilateral problems" P.G. Giarlet (ed.) J.L. Lions (ed.) , Solid Mechanics , Handbook Numer. Anal. , IV , Elsevier (1996) pp. 313–477
[a10]	V.K. Le, K. Schmitt, "Global bifurcation in variational inequalities" , Springer (1997)
[a11]	G. Duvaut, J.L. Lions, "Les Inéquations en Mécanique et en Physique" , Dunod (1972)
[a12]	P.D. Panagiotopoulos, "Inequality problems in mechanics and applications. Convex and nonconvex energy functions" , Birkhäuser (1985)
[a13]	V. Barbu, "Optimal control of variational inequalities" , Res. Notes Math. , 100 , Pitman (1984)