We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N) spaces (X, d, ℋ$$). Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov-Hausdorff limits of the ambient space X along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets E$$ ⊂ X$$ with uniformly bounded measure and perimeter, where (X$$, d$$, ℋ$$) is an arbitrary sequence of RCD(K, N) spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.
Keywords: Isoperimetric problem, existence, direct method, Ricci lower bounds, RCD spaces
@article{COCV_2022__28_1_A57_0,
author = {Antonelli, Gioacchino and Nardulli, Stefano and Pozzetta, Marco},
title = {The isoperimetric problem \protect\emph{via} direct method in noncompact metric measure spaces with lower {Ricci} bounds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022052},
mrnumber = {4467099},
zbl = {1498.53060},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022052/}
}
TY - JOUR AU - Antonelli, Gioacchino AU - Nardulli, Stefano AU - Pozzetta, Marco TI - The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022052/ DO - 10.1051/cocv/2022052 LA - en ID - COCV_2022__28_1_A57_0 ER -
%0 Journal Article %A Antonelli, Gioacchino %A Nardulli, Stefano %A Pozzetta, Marco %T The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022052/ %R 10.1051/cocv/2022052 %G en %F COCV_2022__28_1_A57_0
Antonelli, Gioacchino; Nardulli, Stefano; Pozzetta, Marco. The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 57. doi: 10.1051/cocv/2022052
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