Constrained optimal rearrangement problem leading to a new type obstacle problem

2016/06/10 by Hayk Mikayelyan, Mikayelyan, Hayk
Mathematics · #35R35 #49J20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1606.03174

openalex publication_date 2016/06/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider a new type of obstacle problem in the cylindrical domain Ω=D× (0,1) arising from minimization of the functional ∫_Ω(1)/(2)|∇ u|²+χ_\vgt;0\udx, where v(x')=∫₀¹ 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.

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