We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. The existence is proved in the case of surfaces, and reduced to a topological condition in general. We also construct counter-examples in dimension and to the -regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index of our sequence of critical points converging towards a closed non-trivial geodesic.
Keywords: Geodesics, minimax problems, Finsler geometry
Michelat, Alexis 1 ; Rivière, Tristan 1
@article{COCV_2016__22_4_1282_0,
author = {Michelat, Alexis and Rivi\`ere, Tristan},
title = {A {Viscosity} method for the min-max construction of closed geodesics},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1282--1324},
year = {2016},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
doi = {10.1051/cocv/2016039},
zbl = {1353.49006},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016039/}
}
TY - JOUR AU - Michelat, Alexis AU - Rivière, Tristan TI - A Viscosity method for the min-max construction of closed geodesics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1282 EP - 1324 VL - 22 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016039/ DO - 10.1051/cocv/2016039 LA - en ID - COCV_2016__22_4_1282_0 ER -
%0 Journal Article %A Michelat, Alexis %A Rivière, Tristan %T A Viscosity method for the min-max construction of closed geodesics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1282-1324 %V 22 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016039/ %R 10.1051/cocv/2016039 %G en %F COCV_2016__22_4_1282_0
Michelat, Alexis; Rivière, Tristan. A Viscosity method for the min-max construction of closed geodesics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1282-1324. doi: 10.1051/cocv/2016039
Cité par Sources :