Shape derivatives for minima of integral functionals

Abstract

For $\Omega$ varying among open bounded sets in ${\mathbb R} ^n$, we consider shape functionals $J (\Omega)$ defined as the infimum over a Sobolev space of an integral energy of the kind $\int _\Omega[ f (\nabla u) + g (u) ]$, under Dirichlet or Neumann conditions on $\partial \Omega$. Under fairly weak assumptions on the integrands $f$ and $g$, we prove that, when a given domain $\Omega$ is deformed into a one-parameter family of domains $\Omega _\varepsilon$ through an initial velocity field $V\in W ^ {1, \infty} ({\mathbb R} ^n, {\mathbb R} ^n)$, the corresponding shape derivative of $J$ at $\Omega$ in the direction of $V$ exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of $V$ on $\partial \Omega$. Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.Comment: Mathematical Programming, September 201

On Blaschke-Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue

“…This section is devoted to the proof of Proposition 2.7, namely to the computation of the second order shape derivatives of T and λ 1 at B in dimension 2, with respect to deformations which preserve convexity and keep the volume unchanged. For the formulas of shape derivatives see [19,Chapter 5] and [24,9,10]. Similar computations in terms of Fourier coefficients can be found in [8,1].…”

A Variational Method for Second Order Shape Derivatives

“…Reformulation from a variable domain to a fixed domain. Following the same procedure adopted in [8,Lemma 4.1] in order to recast the first order shape derivative J ′ (Ω, V ), we rewrite the variational problem J(Ω ε ) over the fixed domain Ω: by using a standard change of variables, we obtain…”

“…where u is the unique solution in W 1,p 0 (Ω) to −∆ p u = λ (see for instance [8,14]). It is well-known that u is of class C 1,α (Ω) (see [18,43]), and it is also in C 2 (Ω \ S), where S is the critical set S := {x ∈ Ω : ∇u = 0}.…”

“…is intended as the duality bracket between H −1/2 (∂Ω) and H 1/2 (∂Ω). For a detailed account on the theory of weak traces, we refer to [3,12] (see also [8,Section 2.1]).…”

“…the complement ∂Ω \ Γ D , will be denoted by Γ N . As a natural continuation of our previous paper [8], where we introduced a new approach to first order shape derivatives for functionals of the type (1.1), here we tackle second order shape derivatives by the same method. The main results obtained in [8] are briefly recalled in Section 2, see eq.…”