[Paper Review] Structure of shape derivatives around irregular domains and applications

Jimmy Lamboley, Michel Pierre|arXiv (Cornell University)|Sep 19, 2006

Advanced Mathematical Modeling in Engineering11 references12 citations

TL;DR

This paper establishes a general structure for shape derivatives around sets of finite perimeter in R^N, showing that the derivative depends only on the normal component of deformation at the reduced boundary. It introduces a notion of positivity for shape derivatives that implies continuity with respect to the uniform norm when the boundary is Lipschitz, extending Hadamard's classical result to irregular domains and enabling applications to perimeter and Dirichlet energy functionals.

ABSTRACT

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

Motivation & Objective

• To extend Hadamard's structure theorem for shape derivatives beyond C^2 domains to sets of finite perimeter.
• To define a meaningful notion of positivity for shape derivatives in the context of irregular shapes.
• To establish conditions under which positivity of the shape derivative implies continuity with respect to the uniform norm.
• To apply the framework to key functionals such as perimeter and Dirichlet energy on convex or pseudo-convex sets.
• To demonstrate the sharpness of the Lipschitz boundary condition via a counterexample with a cusp.

Proposed method

• Use geometric measure theory to define the reduced boundary ∂*E and the unit normal ν_E for sets of finite perimeter.
• Define the deformation map Φ: ξ ↦ ξ|Γ · ν_E, mapping vector fields to their normal trace on the reduced boundary Γ.
• Equip the image space Φ(Θ) with a Banach norm to ensure the shape derivative ℒ_E′(0) is represented as a continuous linear form l on Φ(Θ).
• Introduce a notion of positivity for the shape derivative: if ξ·ν_E ≥ 0 H^{N-1}-a.e. on Γ, then ℒ_E′(0)·ξ ≥ 0.
• Prove that positivity of l implies l is continuous with respect to the L^∞(Γ) norm (i.e., uniform norm) when ∂E is Lipschitz.
• Construct a counterexample with a cusp to show that positivity does not imply uniform continuity when the boundary is not Lipschitz.

Experimental results

Research questions

• RQ1 Can the structure of shape derivatives be generalized from C^2 domains to sets of finite perimeter?
• RQ2 What is the appropriate functional analytic framework for shape derivatives when the boundary is irregular?
• RQ3 Does positivity of the shape derivative (in the sense of non-decreasing functionals) imply continuity with respect to the uniform norm?
• RQ4 Is the Lipschitz regularity of the boundary optimal for ensuring such continuity?
• RQ5 Can the theory be applied to perimeter-type and Dirichlet energy functionals on non-smooth convex or pseudo-convex domains?

Key findings

• The shape derivative at a set E of finite perimeter is fully determined by its action on the normal component of deformations at the reduced boundary ∂*E.
• The derivative can be represented as a continuous linear form l on the Banach space Φ(Θ), which is stronger than L^∞(Γ).
• Positivity of the shape derivative—defined as non-negativity when deformations are outward normal—implies continuity with respect to the uniform norm if ∂E is Lipschitz.
• The Lipschitz condition is essentially optimal: a counterexample with a cusp shows that positivity does not imply uniform continuity in the absence of Lipschitz regularity.
• The framework applies to perimeter-type functionals and the Dirichlet energy, providing a rigorous foundation for shape optimization in irregular settings.
• The paper establishes that the total variation measure |∇χ_E| coincides with the (N-1)-dimensional Hausdorff measure restricted to ∂*E.