# A free boundary problem arising in PDE optimization

created by velichkov on 26 May 2015
modified on 21 Apr 2018

[BibTeX]

Published Paper

Inserted: 26 may 2015
Last Updated: 21 apr 2018

Journal: Calc. Var. PDE
Year: 2015

Abstract:

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form

$\displaystyle\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}\vert \nabla u\vert ^2-fu\Big)\,dx.$

We prove the existence of an optimal reinforcement $\theta$ and that it has some higher integrability properties. We also provide some numerical computations for $\theta$ and $u$.