CHAPTER 2 - THE FUNDAMENTAL NOTIONS

Abstract

This chapter focuses on problems in two dimensions, but the approach used is of quite a general character. The properties of the domains considered in the chapter play a fundamental role as far as the behavior of the solutions of partial differential equations are concerned. The Lipschitzian domain is the most general domain dealt with in the chapter. The chapter defines the Sobolev spaces for a bounded domain Ω the definition is also valid for Ω = R2. Some basic well known properties of the Sobolev spaces are reviewed. The applications of n-width in the theory of numerical methods have received considerable attention in recent years. The chapter discusses the n-width for some special spaces, and the weighted Sobolev spaces with weight given by the distance from the boundary. The weighted Sobolev spaces with weight given by the distance from isolated points to the boundary are also discussed.