Shape optimization solutions via Monge-Kantorovich equation. (English. Abridged French version) Zbl 0884.49023
Summary: We consider the optimization problem
\[
\max\{{\mathcal E}(\mu):\mu\text{ nonnegative measure},\;\int d\mu=m\},
\]
where \({\mathcal E}(\mu)\) is the energy associated to \(\mu\):
\[
{\mathcal E}(\mu)= \inf\Biggl\{{1\over 2} \int|Du|^2 d\mu-\langle f,u\rangle: u\in{\mathcal D}(\mathbb{R}^n)\Biggr\}.
\]
The datum \(f\) is a signed measure with finite total variation and zero average. We show that the optimization problem above admits a solution which is not in \(L^1(\mathbb{R}^n)\) in general. This solution comes out by solving a suitable Monge-Kantorovich equation.
MSC:
49Q10 | Optimization of shapes other than minimal surfaces |