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Posted online: 2018-06-15 11:18:06Z by Giuseppe Buttazzo
Cite as: P-180615.1
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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 $$F(\Omega)=\min\left\{\int_\Omega\Big[\frac1p|\nabla u|^p-f(x)u\Big]\,dx\ :\ u\in W^{1,p}_0(\Omega)\right\}$$ 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. On the other hand, if $f\in L^1$, we have counterexamples, i.e. optimal domains are merely quasi-open.
What happens when $f\in L^q$ with $1< q\le d/p$?
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