Asymptotic Solutions in Free Boundary Problems of Singularly Perturbed Elliptic Variational Inequalities

Abstract

This chapter discusses the asymptotic solutions in free boundary problems of singularly perturbed elliptic variational inequalities. The theory of elliptic variational inequalities, which is due to Lions and Stampacchia, is modeled after the variational theory of elliptic boundary value problems; however, its scope is much wider. The variational theory of elliptic boundary value problems leads, in a natural way, to an elliptic variational problem, solutions of which are called weak solutions of the corresponding boundary value problem. An elliptic variational inequality of unilateral type can sometimes be translated into a “differential inequality” such that the (unique) solution of the variational inequality is a solution in a distributional sense of this “differential inequality.” In order to bring out clearly the ideas and the method of analysis, this chapter discusses some simple but representative problems in R . Some elementary results on the behavior of solutions are derived. Elementary properties of solutions and the case of symmetric bilinear forms are discussed in the chapter.