OPTIMAL SHAPES FOR GENERAL INTEGRAL FUNCTIONALS

Problem's Description

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega$ that varies over all subdomains of a given bounded domain $D$ of $R^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur. Consider now the shape optimization problem $$\min\Big\{F(\Omega)\ :\ \Omega\subset D\Big\}$$ with $$F(\Omega)=\min\left\{\int_\Omega\Big[\frac1p|\nabla u|^p-f(x)u\Big]\,dx\ :\ u\in W^{1,p}_0(\Omega)\right\}+|\Omega|$$ and with $f\in W^{-1,p'}$. We know that an optimal domain, in the class of $p$ quasi-open sets, always exists and, when $f\in L^q$ with $q>d/p$, it is an open set. Two questions arise:

(Q1) Is there a counterexample when $f\in L^{d/p}$ to the fact that an optimal domain is open?

(Q2) Do we have additional regularity for optimal domains if we increase the summability of $f$? For instance when $f\in L^q$ with $q>d$?

1. Article Optimal shapes for general integral functionals

   • Giuseppe Buttazzo,
   • Harish Shrivastava

   arXiv