Cylindrical optimal rearrangement problem leading to a new type obstacle problem

Mikayelyan, Hayk¹

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 \Omega =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

$$\begin{array}{c}{\Delta }_u(x\text{'},x_n)={\chi }_{\{v>0\}}(x\text{'})+{\chi }_{\{v=0\}}(x\text{'})\left[{\partial }_{\nu }u(x\text{'},0)+{\partial }_{\nu }u(x\text{'},1)\right]\hfill \end{array}l$$

arising from minimization of the functional

\begin{array}{c}\hfill {\int }_{\Omega }\frac{1}{2}{\left|\nabla u\left(x\right)\right|}^2+{\chi }_{\{v>0\}}\left(x^{\text{'}}\right)u\left(x\right)\mathrm{d}x,\end{array}

where v(x\text{'})={\int }_0^{1}u(x\text{'},t)\mathrm{d}t,\mathrm{and}{\partial }_{\nu }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.

Détail

DOI : 10.1051/cocv/2017047

Classification : 35R35, 49J20
Keywords: Obstacle problem, rearrangements