no. 15-16
Mathematical Analysis/Partial Differential Equations
On a price formation free boundary model by Lasry and Lions: The Neumann problem
[Sur un modèle de formation des prix à frontière libre introduit par Lasry et Lions : Le problème de Neumann]

Présenté par : Pierre-Louis Lions

Caffarelli, Luis A.1 ; Markowich, Peter A.2, 3 ; Wolfram, Marie-Therese3
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. 15-16, pp. 841-844.

Nous discutons lʼexistence locale et globale, ainsi que lʼunicité des solutions pour le modèle de formation des prix à frontière libre avec des conditions aux bords de Neumann homogènes introduit par Lasry et Lions en 2007. Nos résultats sont basés sur une transformation de ce problème en une équation de la chaleur avec des conditions aux bords non standard. La frontière libre devient la ligne de niveau zéro de la solution de lʼéquation de la chaleur. Cette transformation nous permet de construire une solution explicite et de discuter le comportement de la frontière libre. Lʼexistence globale peut être vérifiée sous certaines conditions sur la frontière libre, et nous donnons des exemples de non-existence.

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry and Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.006
Caffarelli, Luis A. 1 ; Markowich, Peter A. 2, 3 ; Wolfram, Marie-Therese 3

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_15-16_841_0,
     author = {Caffarelli, Luis A. and Markowich, Peter A. and Wolfram, Marie-Therese},
     title = {On a price formation free boundary model by {Lasry} and {Lions:} {The} {Neumann} problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {841--844},
     publisher = {Elsevier},
     volume = {349},
     number = {15-16},
     year = {2011},
     doi = {10.1016/j.crma.2011.07.006},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.07.006/}
}
TY  - JOUR
AU  - Caffarelli, Luis A.
AU  - Markowich, Peter A.
AU  - Wolfram, Marie-Therese
TI  - On a price formation free boundary model by Lasry and Lions: The Neumann problem
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 841
EP  - 844
VL  - 349
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.07.006/
DO  - 10.1016/j.crma.2011.07.006
LA  - en
ID  - CRMATH_2011__349_15-16_841_0
ER  - 
%0 Journal Article
%A Caffarelli, Luis A.
%A Markowich, Peter A.
%A Wolfram, Marie-Therese
%T On a price formation free boundary model by Lasry and Lions: The Neumann problem
%J Comptes Rendus. Mathématique
%D 2011
%P 841-844
%V 349
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.07.006/
%R 10.1016/j.crma.2011.07.006
%G en
%F CRMATH_2011__349_15-16_841_0
Caffarelli, Luis A.; Markowich, Peter A.; Wolfram, Marie-Therese. On a price formation free boundary model by Lasry and Lions: The Neumann problem. Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 841-844. doi : 10.1016/j.crma.2011.07.006. http://www.numdam.org/articles/10.1016/j.crma.2011.07.006/

[1] Avdonin, S.A.; Joó, I. Riesz basis of exponentials and sine-type functions, Acta Math. Hungar., Volume 51 (1988) no. 1, pp. 3-14

[2] Caffarelli, L.A.; Markowich, P.A.; Pietschmann, J.-F. On a price formation free boundary problem by Lasry & Lions, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 11–12, pp. 621-624

[3] 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

[4] González, M.; Gualdani, M.P. Asymptotics for a free-boundary problem in price formation, Nonlinear Anal., Volume 74 (2011), pp. 3269-3294

[5] Horváth, M. On the Muckenhoupt condition, Period. Math. Hungar., Volume 18 (1987) no. 1, pp. 53-58

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

[7] 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 :