ABSTRACT

In this chapter, we recall the concept of shape differential equation developed in [145],[147]. Here, we present a simplified version and some applications in dimension 2 which enable us to reach the time asymptotic result. We consider a shape functional J which is shape differentiable in Ok with respect to Vk to be specified later on. We denote ∇J(Ω) its gradient, considered as a distribution in A∗k. For any Ω0 in Ok and V in Vk, the absolute continuity of J writes

J(Ωs(V))− J(Ω0) = ∫ s

〈∇J(Ωt(V)) , V(t)〉A∗k× Ak dt , ∀ s > 0 (4.1)

Classical gradient based methods allow us to control the variations of J with respect to the domain. Considering the problem,

J(Ω(V))

we would like to elaborate a constructive method to decrease the functional following the gradient. This may be done by solving the non-linear equation for large evolution of the domain,

∇J(Ωt(V)) + A(V(t)) = 0, ∀ t > 0 (4.2)

where A is an ad-hoc duality operator.This corresponds to the well-known steepest descent method. From the structure theorem for shape gradient [135], we have1,

∇J(Ω) = γ∗Γ · (g n)

where the shape density gradient g is a distribution on the boundary Γ. Usually it is a function on Γ so that we consider any extension G of g defined

Γ. In the term ∗Γ · (g n) be identified with G ∇χ. Hence, the shape differential equation turns into an Hamilton-Jacobi equation for the characteristic function χ,{

∂t χ+ 〈∇χ,A−1 · (G(χ)∇χ)〉 = 0, (0, τ) χ(0) = χΩ0 , Ω0 ⊂ D (4.3)

We shall see in the sequel that the previous equation can be weakened using the level set formulation where we shall solve the following equation,{

∂t Φ− 〈∇Φ,A−1 · (G(χt(Φ))∇χt(Φ))〉 = 0, (0, τ) Φ(0) = Φ0,

(4.4)

where χt(Φ) = {x ∈ D, Φ(t, x) > 0}. We are going to recall the constructive proof of the existence of a V satisfying (4.2) and investigate the asymptotic behaviour of the method. The existence of a solution for this so-called shape differential equation has been proven in [147] inside a larger setting2.