On a shape derivative formula for star-shaped domains using Minkowski deformation

Abstract

We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem.

Keywords

• shape derivative
• volume functional
• convex domain
• star-shaped domain
• support function
• gauge function
• minkowski deformation
• shape optimization
• brunn-minkowski theory