Constrained optimal rearrangement problem leading to a new type obstacle problem

Abstract

We consider a new type of obstacle problem in the cylindrical domain $\Omega =D\times (0,1)$ arising from minimization of the functional $${\int }_{\Omega }\frac{1}{2}|\nabla u{|}^2+{\chi }_{\{v>0\}}udx,$$ where $v(x^{\prime })={\int }_0^{1}u(x^{\prime },t)dt$. We prove several existence and regularity results and show that the comparison principle does not hold for minimizers. This problem is derived from a classical optimal rearrangement problem in a cylindrical domain, under the constraint that the force function does not depend on the $x_n$ variable of the cylindrical axis. A mistake in the Theorem 4.2 of the previous version has been found. The statement remains an open problem.