In this chapter we introduce the concept of a variational inequality and establish general existence and uniqueness theorems. Regularity results are proved for some classes of variational inequalities, mainly the obstacle prob- lem, the case of gradient constraint, the biharmonic obstacle problem, and the case of thin obstacles.
We introduce several physical problems to which the existence and regular- ity theorems are applied, such as (i) the filtration of.water in porous medium, (ii) the elastic—plastic torsion problem, and (iii) the Stefan problem of the melting of a solid.
Let be a bounded domain in R”. We denote by the class of functions u(x) in LP(IZ) such that all their weak derivatives Dau =
• • of orders m belong to here a = (a1,.. a1
a1 + •. +a,,, D1 a/ax1, and the weak derivative is defined by
fDttu •(x)dx = (— V 4) E
stands for functions with compact support in is a Banach space with the norm
here I < oo; forp oo we define
I U ess sup I Dau I
where D°u are taken as weak derivatives. It is well known (see references 94c and 109) that
coc(a) n is dense in
The closure of in is denoted by The notation
= WmPffl), Hm(12) = = W”2(1l)
is also customary. Consider the functional
and the closed convex set in H1(tZ).
(1.2) K=
wheref is a in is a continuous function in ft and g E H'(fl). We assume that g 4′; then K is nonempty.
Consider the problem: Finsi u such that
(1.3) u — g E G(u) mm G(v).
Suppose that u isa solution of this problem. Then
and we easily deduce that
Hence, if u E H2((2),
so that
(1.4) inft
This equation is satisfied, of course, in the a.e. sense; the condition u — g E is a generalized version of the Dmchlet boundary condition (see references 94c and 109) (1.5) ug onaf�. Thus the solution of the minimization problem (1.3) is also the solution of the Dirichiet problem (1.4), (1.5). Consider next the variational problem: Find u such that (1.6) u€K, G(u)=minG(v). vEK

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