Cylindrical optimal rearrangement problem leading to a new type obstacle problem

Mikayelyan, Hayk

doi:10.1051/cocv/2017047

Mikayelyan, Hayk ¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 859-872.

Résumé

An optimal rearrangement problem in a cylindrical domain $Ω = D \times (0, 1)$ 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

\begin{matrix} Δ_{u} (x', x_{n}) = χ_{{v > 0}} (x') + χ_{{v = 0}} (x') [\partial_{ν} u (x', 0) + \partial_{ν} u (x', 1)] \end{matrix} l

arising from minimization of the functional

\begin{matrix} \int_{Ω} \frac{1}{2} {|\nabla u (x)|}^{2} + χ_{{v > 0}} (x^{'}) u (x) d x, \end{matrix}

where $v (x') = \int_{0}^{1} u (x', t) d t, and$ $\partial_{ν} u$ 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.

MR Zbl

DOI : 10.1051/cocv/2017047

Classification : 35R35, 49J20
Mots clés : Obstacle problem, rearrangements

Affiliations des auteurs :

Mikayelyan, Hayk ¹

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     author = {Mikayelyan, Hayk},
     title = {Cylindrical optimal rearrangement problem leading to a new type obstacle problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {859--872},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {2},
     year = {2018},
     doi = {10.1051/cocv/2017047},
     zbl = {1402.49007},
     mrnumber = {3816419},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017047/}
}

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Mikayelyan, Hayk. Cylindrical optimal rearrangement problem leading to a new type obstacle problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 859-872. doi : 10.1051/cocv/2017047. http://www.numdam.org/articles/10.1051/cocv/2017047/

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