In this paper a rearrangement minimization problem corresponding to solutions of the p-Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in ℝ2 and ℝ3.
Keywords: $$-Laplacian, rearrangement minimization, analytical solution, rearrangement algorithms, membranes
@article{COCV_2022__28_1_A11_0,
author = {Kao, Chiu-Yen and Mohammadi, Seyyed Abbas},
title = {A rearrangement minimization problem corresponding to $p${-Laplacian} equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022004},
mrnumber = {4379605},
zbl = {1492.35392},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022004/}
}
TY - JOUR AU - Kao, Chiu-Yen AU - Mohammadi, Seyyed Abbas TI - A rearrangement minimization problem corresponding to $p$-Laplacian equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022004/ DO - 10.1051/cocv/2022004 LA - en ID - COCV_2022__28_1_A11_0 ER -
%0 Journal Article %A Kao, Chiu-Yen %A Mohammadi, Seyyed Abbas %T A rearrangement minimization problem corresponding to $p$-Laplacian equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022004/ %R 10.1051/cocv/2022004 %G en %F COCV_2022__28_1_A11_0
Kao, Chiu-Yen; Mohammadi, Seyyed Abbas. A rearrangement minimization problem corresponding to $p$-Laplacian equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 11. doi: 10.1051/cocv/2022004
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