We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities.
Nous obtenons un nouveau résultat de stabilité pour les solutions de viscosité d'équations fortement non linéaires paraboliques dans le cas où l'on n'a qu'une convergence faible en temps pour les non-linéarités.
Accepted:
Published online:
Guy Barles 1
@article{CRMATH_2006__343_3_173_0, author = {Guy Barles}, title = {A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time}, journal = {Comptes Rendus. Math\'ematique}, pages = {173--178}, publisher = {Elsevier}, volume = {343}, number = {3}, year = {2006}, doi = {10.1016/j.crma.2006.06.022}, language = {en}, }
TY - JOUR AU - Guy Barles TI - A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time JO - Comptes Rendus. Mathématique PY - 2006 SP - 173 EP - 178 VL - 343 IS - 3 PB - Elsevier DO - 10.1016/j.crma.2006.06.022 LA - en ID - CRMATH_2006__343_3_173_0 ER -
Guy Barles. A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 173-178. doi : 10.1016/j.crma.2006.06.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.022/
[1] Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997
[2] A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., Volume 32 (1995) no. 2, pp. 484-500
[3] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions, Preprint
[4] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions, Existence and applications to the level-set approach, Preprint
[5] User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 1, pp. 1-67
[6] Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics (New York), vol. 25, Springer-Verlag, New York, 1993
[7] Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., Volume 28 (1985), pp. 33-77
[8] Remarks on Hamilton–Jacobi equations with measurable time-dependent Hamiltonians, Non-linear Analysis. Theory Methods and Applications, Volume 11 (1987), pp. 613-621
[9] Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998) no. 9, pp. 1085-1092
[10] Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math., Volume 327 (1998) no. 8, pp. 735-741
[11] P.L. Lions, P.E. Souganidis, personal communications, various courses and lectures
[12] Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence, Differential and Integral Equations, Volume 3 (1990) no. 1, pp. 77-91
[13] Existence and uniqueness of viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Analysis. Theory, Methods and Applications, Volume 18 (1992) no. 11, pp. 1033-1062
Cited by Sources:
Comments - Policy