Voir la notice de l'article provenant de la source Numdam
We consider the diffusive Hamilton–Jacobi equation, with superquadratic Hamiltonian, homogeneous Dirichlet conditions and regular initial data. It is known from [4] (Barles–DaLio, 2004) that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now.
Our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary.
@article{AIHPC_2017__34_7_1913_0, author = {Porretta, Alessio and Souplet, Philippe}, title = {Analysis of the loss of boundary conditions for the diffusive {Hamilton{\textendash}Jacobi} equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1913--1923}, publisher = {Elsevier}, volume = {34}, number = {7}, year = {2017}, doi = {10.1016/j.anihpc.2017.02.001}, mrnumber = {3724761}, zbl = {1391.35078}, language = {en}, url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.02.001/} }
TY - JOUR AU - Porretta, Alessio AU - Souplet, Philippe TI - Analysis of the loss of boundary conditions for the diffusive Hamilton–Jacobi equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1913 EP - 1923 VL - 34 IS - 7 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.02.001/ DO - 10.1016/j.anihpc.2017.02.001 LA - en ID - AIHPC_2017__34_7_1913_0 ER -
%0 Journal Article %A Porretta, Alessio %A Souplet, Philippe %T Analysis of the loss of boundary conditions for the diffusive Hamilton–Jacobi equation %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1913-1923 %V 34 %N 7 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.02.001/ %R 10.1016/j.anihpc.2017.02.001 %G en %F AIHPC_2017__34_7_1913_0
Porretta, Alessio; Souplet, Philippe. Analysis of the loss of boundary conditions for the diffusive Hamilton–Jacobi equation. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1913-1923. doi: 10.1016/j.anihpc.2017.02.001
Cité par Sources :