Shape flows for spectral optimization problems (Q1947210)

The authors study some problems arising in spectral optimization. These optimization problems can be written as minimum problems like \(\min F(\Omega):\;\Omega\in \mathfrak{A}\) where \(\mathfrak{A}\) is a suitable family of admissible domains and \(F\) is a suitable cost function defined on \(\mathfrak{A}\). The admissible class \(\mathfrak{A}\) is made from domains in \(\mathbb{R}^d\) and the cost functional is of the type of integral functional or spectral functional. The authors study the shape evolution \(\Omega(t)\), starting form a given domain \(\Omega_0\) according to a suitable definition of gradient flow. First, the authors study the problem of the existence of a relaxed flow. The problem is presented in terms of flow capacitary measures and exploit a specific one-to-one relation to flows in convex sets in \(L^2\). The case where the flow is made of relaxed domains is studied. Some examples show that even starting from a very smooth initial domain \(\Omega_0\), the gradient flow quits the admissible class \(\mathfrak{A}\) to evolve in the class of relaxed shapes. Next, the authors show that, in some situations, one can obtain an evolution \(\Omega(t)\) consisting of classical domains. Such a situation can appear in the case where the functionals \(F\) are monotone by set inclusion. A general existence result and some properties of the path \(\Omega (t)\), as well as some open problems are presented at the end of the paper.