We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.
Keywords: front propagation, non-local equations, dislocation dynamics, mean curvature motion, viscosity solutions, minimizing movements, sets of finite perimeter, currents
@article{COCV_2009__15_1_214_0,
author = {Forcadel, Nicolas and Monteillet, Aur\'elien},
title = {Minimizing movements for dislocation dynamics with a mean curvature term},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {214--244},
year = {2009},
publisher = {EDP-Sciences},
volume = {15},
number = {1},
doi = {10.1051/cocv:2008027},
mrnumber = {2488577},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008027/}
}
TY - JOUR AU - Forcadel, Nicolas AU - Monteillet, Aurélien TI - Minimizing movements for dislocation dynamics with a mean curvature term JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 214 EP - 244 VL - 15 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008027/ DO - 10.1051/cocv:2008027 LA - en ID - COCV_2009__15_1_214_0 ER -
%0 Journal Article %A Forcadel, Nicolas %A Monteillet, Aurélien %T Minimizing movements for dislocation dynamics with a mean curvature term %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 214-244 %V 15 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2008027/ %R 10.1051/cocv:2008027 %G en %F COCV_2009__15_1_214_0
Forcadel, Nicolas; Monteillet, Aurélien. Minimizing movements for dislocation dynamics with a mean curvature term. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 214-244. doi: 10.1051/cocv:2008027
Cité par Sources :