ARIATIONAL INEQUALITIES: EXISTENCE AND REGULARITY

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.

1 AN EXAMPLE.

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

‘I

I

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.

1 AN EXAMPLE.

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

‘I

I

AN EXAMPLE

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

(1.1)

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

ereal,

and we easily deduce that

Hence, if u E H2((2),

so that

(1.4) inft

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

(1.1)

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

ereal,

and we easily deduce that

Hence, if u E H2((2),

so that

(1.4) inft

Suppose that u isa solution of this problem. Then ereal, and we easily deduce that Hence, if u E H2((2), so that (1.4) inft