Preliminaries and the material derivative method

Abstract

This chapter is concerned with mathematical methods used in the shape sensitivity analysis. In particular in Sect. 2.9 the so-called material derivative and the speed method are introduced. The latter is applied in Chaps. 3 and 4 for the shape sensitivity analysis of the boundary value problems of elliptic type as well as for the initial — boundary value problems of parabolic and hyperbolic types. In Chap. 4 the speed method is used for the shape sensitivity analysis of nonlinear problems of elliptic type. In Sect. 3.3 of Chap. 3, the necessary optimality conditions for a model shape optimization problem are derived using the speed method. In this chapter we describe mathematical tools that can be used to prove the existence of solutions to related shape optimization problems, e.g. the notion of the perimeter of a bounded domain in ℝN (see (DeGiorgi et al. 1972)) is discussed. In Sect. 2.7 we introduce, following (Micheletti 1972), the notion of the convergence of domains that ensures e.g. the convergence of normal vector fields on the boundaries, as well as the curvatures of the boundaries, etc. In Sect. 2.1 the domain Ω ⊂ ℝN with the boundary Γ = ∂ Ω is defined. The notion of an integral on the manifold Γ = ∂ Ω is discussed in Sect. 2.2. Functional spaces used in the book are examined in Sect. 2.3, in particular the Sobolev spaces (see e.g. (Adams 1975; Lions et al. 1968)) are introduced. In Sect. 2.4 the notion of weak solutions to elliptic boundary value problems is investigated using Stampacchia’s theorem, and the Lax-Milgram lemma in the symmetric case.