Minimization problem. Variational inequalities

Let V be a normed space, and J V — i be an arbitrary functional. We assume that there exists a linear and continuous functional such that for each u G V 

It is said in this case that the functional J has the derivative at the point u. Let V be the space dual of V, i.e. the space of all linear continuous functionals on V. If the operator J V — V is defined such that for each u gV the derivative can be found at the point u, then the functional J is called differentiable. 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives 

This exactly coincides with the Euler equation. 

Let the functional J be convex and differentiable. We can prove the validity of the inequality 

---

Let us emphasize that not model can be presented as a minimization problem like (1.55) or (1.57). Thus, elastoplastic problems considered in Chapter 5 can be formulated as variational inequalities, but we do not consider any minimization problems in plasticity. In all cases, we have to study variational problems or variational inequalities. It is a principal topic of the following two sections. As for general variational principles in mechanics and physics we refer the reader to (Washizu, 1968 Chernous ko, Banichuk, 1973 Ekeland, Temam, 1976 Telega, 1987 Panagiotopoulos, 1985 Morel, Solimini, 1995). 

The equilibrium problem for the plate can be formulated as variational, namely, it corresponds to the minimum of the functional H over the set of admissible displacements. To minimize the functional H over the set we can consider the variational inequality... 

Herewith the problem of minimizing H over the set is equivalent to the variational inequality... 

The considered problem is formulated as a variational inequality. In general, the equations (3.140)-(3.142) hold in the sense of distributions. In addition to (3.143), complementary boundary conditions will be fulfilled on F, X (0,T). The exact form of these conditions is given at the end of the section. The assumption as to sufficient solution regularity requires the variational inequality to be a corollary of (3.140)-(3.142), the initial and all boundary conditions. The relationship between these two problem formulations is discussed in Section 3.4.4. We prove an existence of the solution in Section 3.4.2. In Section 3.4.3 the main result of the section concerned with the cracks of minimal opening is established. 

---