Structure of shape derivatives around irregular domains and applications

2006/09/19 by Jimmy Lamboley, Lamboley, Jimmy, Michel Pierre +1
Computer Science · Engineering · #49J50 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical methods in engineering #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.math/0609526

openalex publication_date 2006/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Estimates of First and Second Order Shape Derivatives in Nonsmooth Multidimensional Domains and Applications
• Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
• Optimal shapes for general integral functionals
• A structure theorem for shape functions defined on submanifolds
• Regularity of Minimizers of Shape Optimization Problems involving Perimeter