$C^2,\alpha$ Existence Result for a Class of Shape Optimization Problems
Arian Novruzi · SIAM Journal on Control and Optimization · 2004
In this paper we give a sufficient condition for the existence of regular optimal domains for a class of shape optimization problems. This consists of finding a $C^{2,\alpha}$ domain minimizing locally a shape functional depending on the perimeter of the domain and on a general term, which in most PDE applications represents the energy associated with the state equation, under the constraint that the measure of the domain is given. The proof of this result is based on another existence result for $C^{2,\alpha}$ solutions for a class of free boundary problems that are critical domains for the shape functionals considered previously. A key point is the introduction of a special domain transformation, which has a separate term responsible for the domain translation and another which is basically only responsible for "pure" deformation of the domain. As an application, a typical example involving the Dirichlet problem in R N is considered.