Constrained optimal rearrangement problem leading to a new type obstacle problem

Hayk Mikayelyan

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arXiv · 1606.03174

Constrained optimal rearrangement problem leading to a new type obstacle problem

Hayk Mikayelyan

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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.

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Hayk Mikayelyan. 2016-06-10. Constrained optimal rearrangement problem leading to a new type obstacle problem. https://arxiv.org/abs/1606.03174

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