Cylindrical optimal rearrangement problem leading to a new type obstacle problem

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 _Ω1/2|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.