On the normal variations of a domain

Abstract

In domain optimization problems, normal variations of a reference domain are frequently used. We prove that such variations do notpreserve the regularity of the domain. More precisely, we give a bounded domain which boundary is m times differentiable and ascalar variation which is infinitely differentiable such that the deformed boundary is only m-1 times differentiable. We prove inaddition that the only normal variations which preserve the regularity are those with constant magnitude. This shows that the use of normal variations in an iterative approximation method for domain optimization generates a loss ofregularity at each iteration, and thus it is better to use transverse variations which preserve the regularity of the domain.