A free boundary problem arising in PDE optimization

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 $$\underset{{\int }_D\theta dx=m}{\sup }\underset{u\in H_0^1(D)}{\inf }{\int }_D(\frac{1+\theta }{2}|\nabla u{|}^2-fu)dx.$$ 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 $u$.

29 pages, 42 figures