no. 11-12
Mathematical Analysis/Partial Differential Equations
On a price formation free boundary model by Lasry and Lions
[Sur un modèle de formation de prix de Lasry et Lions]

Présenté par : Pierre-Louis Lions

Caffarelli, Luis A.1 ; Markowich, Peter A.2, 3 ; Pietschmann, Jan-F.2
1 Department of Mathematics, Institute for Computational Engineering and Sciences, University of Texas at Austin, USA
2 DAMTP, University of Cambridge, Cambridge CB3 0WA, UK
3 Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 621-624.

Nous discutons lʼexistence globale et le comportement local dʼun modèle de formation de prix introduit par Lasry et Lions en 2007. Nos résultats reposent sur une transformation reliant ce problème à une solution de lʼéquation de la chaleur avec données initiales particulières. Le point clé est que la frontière libre, présente dans le problème original, est conservée comme ensemble de niveau zéro de cette solution. Utilisant les propriétés de lʼopérateur de la chaleur nous pouvons montrer des résultats de régularité de la frontière libre et par suite lʼexistence globale.

We discuss global existence and asymptotic behaviour of a price formation free boundary model introduced by Lasry and Lions in 2007. Our results are based on a construction which transforms the problem into the heat equation with specially prepared initial datum. The key point is that the free boundary present in the original problem becomes the zero level set of this solution. Using the properties of the heat operator we can show global existence, regularity and asymptotic results of the free boundary.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.05.011
Caffarelli, Luis A. 1 ; Markowich, Peter A. 2, 3 ; Pietschmann, Jan-F. 2

1 Department of Mathematics, Institute for Computational Engineering and Sciences, University of Texas at Austin, USA
2 DAMTP, University of Cambridge, Cambridge CB3 0WA, UK
3 Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria 
@article{CRMATH_2011__349_11-12_621_0,
     author = {Caffarelli, Luis A. and Markowich, Peter A. and Pietschmann, Jan-F.},
     title = {On a price formation free boundary model by {Lasry} and {Lions}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {621--624},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.05.011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.05.011/}
}
TY  - JOUR
AU  - Caffarelli, Luis A.
AU  - Markowich, Peter A.
AU  - Pietschmann, Jan-F.
TI  - On a price formation free boundary model by Lasry and Lions
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 621
EP  - 624
VL  - 349
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.05.011/
DO  - 10.1016/j.crma.2011.05.011
LA  - en
ID  - CRMATH_2011__349_11-12_621_0
ER  - 
%0 Journal Article
%A Caffarelli, Luis A.
%A Markowich, Peter A.
%A Pietschmann, Jan-F.
%T On a price formation free boundary model by Lasry and Lions
%J Comptes Rendus. Mathématique
%D 2011
%P 621-624
%V 349
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.05.011/
%R 10.1016/j.crma.2011.05.011
%G en
%F CRMATH_2011__349_11-12_621_0
Caffarelli, Luis A.; Markowich, Peter A.; Pietschmann, Jan-F. On a price formation free boundary model by Lasry and Lions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 621-624. doi : 10.1016/j.crma.2011.05.011. http://www.numdam.org/articles/10.1016/j.crma.2011.05.011/

[1] Chayes, L.; González, M.; Gualdani, M.P.; Kim, I. Global existence and uniqueness of solutions to a model of price formation, SIAM J. Math. Anal., Volume 41 (2009) no. 5, pp. 2107-2135

[2] González, M.; Gualdani, M.P. Asymptotics for a free-boundary problem in price formation, Nonlinear Anal., Theory Methods Appl., Volume 74 (2011) no. 10, pp. 3269-3294

[3] Lasry, J.-M.; Lions, P.-L. Mean field games, Jpn. J. Math., Volume 2 (2007) no. 1, pp. 229-260

[4] Markowich, P.A.; Matevosyan, N.; Pietschmann, J.-F.; Wolfram, M.-T. On a parabolic free boundary equation modeling price formation, Math. Models Methods Appl. Sci., Volume 11 (2009) no. 19, pp. 1929-1957

Cité par Sources :