Cylindrical optimal rearrangement problem leading to a new type obstacle problem^★

School of Mathematical Sciences, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo 315100, Zhejiang Prov., P.R. China

^a Corresponding author: Hayk.Mikayelyan@nottingham.edu.cn

Received: 22 August 2016
Revised: 13 April 2017
Accepted: 15 June 2017

Abstract

An optimal rearrangement problem in a cylindrical domain Ω = D × (0, 1) is considered, under the constraint that the force function does not depend on the x_n variable of the cylindrical axis. This leads to a new type of obstacle problem in the cylindrical domain

     Δu(x′, x_n) = χ_{v>0} (x′) + χ_{v=0}(x′) [∂_νu(x′,0) + ∂_νu(x′, 1)]

arising from minimization of the functional

     ∫_Ω½;|∇u(x)|² + χ_{v>0} (x′)u(x) dx,

where v(x′) = ∫₀¹ u(x′, t)dt, and ∂_νu is the exterior normal derivative of u at the boundary. Several existence and regularity results are proven and it is shown that the comparison principle does not hold for minimizers.