A free boundary problem arising in PDE optimization

TL;DR

This paper studies a free boundary problem from PDE optimization related to membrane reinforcement and traffic congestion, proving existence of optimal solutions, their properties, and providing numerical examples.

Contribution

It establishes the existence and higher integrability of optimal reinforcement functions and solutions, along with numerical computations for the problem.

Findings

Existence of an optimal reinforcement function $ heta$

Higher integrability properties of the optimal solutions

Numerical computations illustrating the optimal reinforcement and solution

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