Inner Product of Functions. Norm, Metric

Abstract

To start with, a few comments concerning the notation employed: By the symbol G we denote — throughout the book — an N-dimensional domain, i.e., an open connected set in the Euclidean space E _N . Only bounded domains with the so-called Lipschitz boundary will be considered in this book. The definition of this concept is rather complicated and would cause the reader unnecessary difficulties at this opening stage. Therefore, it is left over to Chap. 28. Here we note only that the definition is sufficiently general to include the domains most frequently encountered in engineering applications as far as the solution of boundary value problems in differential equations of the elliptic type is in question. For instance, to this kind of domains there belong plane domains with a smooth or piecewise smooth boundary having no cuspidal points; in the three-dimensional space domains whose boundary is smooth or piecewise smooth and has no singularities corresponding, in a certain sense, to cuspidal points of plane curves (cuspidal edges, etc.). Examples of plane and space domains (i.e., for N = 2 and N = 3) with the Lipschitz boundary are circles, annuli, squares, triangles, spheres, cubes, etc. For N = 1, the domain reduces to an interval (a, b). Whenever more than one domain is considered subscripts are used to distinguish them: G₁G₂, etc.