Introduction In this book we treat free boundary problems of the type Δu = f(x, u, ∇u) in D Rn, where the right-hand side f exhibits a jump discontinuity in its second and/or third variables. The discontinuity set is a priori unknown and there- fore is said to be free. The prototypical example is the so-called classical obstacle problem, which minimizes the energy of a stretched membrane over a given obstacle (see more in §1.1) and in its simplest form can be reformu- lated as (0.1) Δu = χ{u0}, u 0, in D. In this case f(x, u, ∇u) = χ{u0} is the Heaviside function of u. The free boundary here is Γ = ∂{u 0}∩D. Also note that the sign condition u 0 in (0.1) appears naturally in this problem. The classical obstacle problem and its variations have been the subject of intense studies in the past few decades. Today, there is a more or less complete and comprehensive theory for this problem, both from theoretical and numerical points of view. The motivation for studying free boundary problems in general, and obstacle-type problems in particular, has roots in many applications. Clas- sical applications of these problems originate (predominantly) in engineer- ing sciences, where many problems (sometimes after a major simplification) could be formulated as variational inequalities or more general free boundary problems. In many cases variational inequalities can be viewed as obstacle- type problems, with an additional sign condition, as in the case of the clas- sical obstacle problem. This particular feature significantly simplifies the problem, and most methods, up to the early 1990s, relied heavily on this strong property of solutions. 1
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