We solve a class of isoperimetric problems on $$ with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β + 1, β ≤ 2α. We show that, among all smooth sets Ω in $$ with fixed weighted measure ∬Ωy$$dxdy, the weighted perimeter ∫$$y$$ ds achieves its minimum for a smooth set which is symmetric w.r.t. to the y-axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a bound for eigenvalues of some nonlinear problems.
Keywords: Isoperimetric inequality, weighted Cheeger set, eigenvalue problems
@article{COCV_2021__27_S1_A4_0,
author = {Alvino, Angelo and Brock, Friedemann and Chiacchio, Francesco and Mercaldo, Anna and Posteraro, Maria Rosaria},
title = {Some isoperimetric inequalities with respect to monomial weights},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020054},
mrnumber = {4222168},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020054/}
}
TY - JOUR AU - Alvino, Angelo AU - Brock, Friedemann AU - Chiacchio, Francesco AU - Mercaldo, Anna AU - Posteraro, Maria Rosaria TI - Some isoperimetric inequalities with respect to monomial weights JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020054/ DO - 10.1051/cocv/2020054 LA - en ID - COCV_2021__27_S1_A4_0 ER -
%0 Journal Article %A Alvino, Angelo %A Brock, Friedemann %A Chiacchio, Francesco %A Mercaldo, Anna %A Posteraro, Maria Rosaria %T Some isoperimetric inequalities with respect to monomial weights %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020054/ %R 10.1051/cocv/2020054 %G en %F COCV_2021__27_S1_A4_0
Alvino, Angelo; Brock, Friedemann; Chiacchio, Francesco; Mercaldo, Anna; Posteraro, Maria Rosaria. Some isoperimetric inequalities with respect to monomial weights. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S3. doi: 10.1051/cocv/2020054
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