Constrained optimal rearrangement problem leading to a new type obstacle problem

TL;DR

This paper introduces a new obstacle problem derived from a constrained optimal rearrangement in cylindrical domains, establishing existence and regularity results, and highlighting unique properties like the failure of the comparison principle.

Contribution

It formulates a novel obstacle problem from a classical rearrangement problem with a non-standard constraint, providing foundational theoretical results and identifying open issues.

Findings

Existence and regularity of solutions are established.

Comparison principle does not hold for minimizers.

Identified open problem related to Theorem 4.2.

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')=\int_0^1 u(x', 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.