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## gradient constraint,

INTRODUcTION

approach, existence, uniqueness. and regularity of the minimizer. Chapter 2 is concerned with the study of the free boundary itself.

In Chapter 3 we study a class of variational problems designed for solving problems of jets and cavities of ideal fluids. Whereas in Chapter 1 a typical functional is

f[$vvI2 + 2fvJ (r>O)

in Chapter 3 the corresponding functional is

where ‘A is the characteristic function of A. The variational functional in Chapter 4 is of the type

+JA(p(x)).

where p is a density function subject to some constraints. Here the free boundary is 8(p > 0).

In Chapter 5 we study several free-boundary problems that are not for- rnulted as variational problems; we deal mainly with gas ;n a porous medium and with the filtration of fluid in a porous dam.

Chapters 3, 4, and 5 are basically independent of each other as far as cross references are concerned. However, they do share some common methodL techniques, and ideas. The material of Chapters 1 and 2 appears in later chapters either directly or indirectly.

There is a large body of literature on time-dependent fke-boundary prob- lems in one space dimension. Here the methods are often highly specialized. With a few exceptions, we shall not deal with such problems in the present book.

1

VARIATIONAL 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 problem