Chapter 1 Model problems Free boundary problems of obstacle type appear naturally in numerous ap- plications, and the purpose of this chapter is to list some of the most inter- esting ones, from our point of view (§1.1). Not all of these problems will be treated in detail in this book, but the methods developed here will be applicable to all of them at least with a certain degree of success. For the detailed treatment we have selected three model problems (which we call A, B, and C) that can be put into a more general framework of obstacle-type problems OT1–OT2 (§1.2). At the end of this chapter (§1.3) we discuss the almost optimal W 2,p loc C 1,α loc regularity of solutions for any 1 p ∞, 0 α 1. 1.1. Catalog of problems 1.1.1. The classical obstacle problem. The Dirichlet principle. A well-known variational principle of Dirichlet says that the solution of the boundary value problem Δu = 0 in D, u = g on ∂D, can be found as the minimizer of the (Dirichlet) functional J0(u) = D |∇u|2dx, among all u such that u = g on ∂D. More precisely (and slightly more generally), if D is a bounded open set in Rn, g W 1,2(D) and f L∞(D), then the minimizer of (1.1) J(u) = D (|∇u|2 + 2fu)dx 7
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