Introduction

Abstract

Throughout this book, Ω $\Omega \subseteq \mathbb{R}^n$ ℝⁿ means a general domain, that is any open nonempty connected subset of the n-dimensional Euclidean space ℝⁿ. In the linearized theory we admit that n ≥ 2, the nonlinear theory is restricted to n = 2 and n = 3; in the preliminaries, see Chapters I and II, we sometimes admit the case n = 1. ∂Ω always means the boundary of Ω.Ω may be unbounded and ∂Ω may be also unbounded. In the sections on regularity properties, we suppose certain smoothness conditions on the boundary ∂Ω. The variables x = (x ₁…x _n ) ∈ Ω are called space variables. T is always given with 0 < T ≤ ∞ and [0,T) is called the time interval; t ∈ [0,T) is called the time variable. We admit the case T = ∞.