Existence of weak solutions to first-order stationary mean-field games with dirichlet conditions

Rita Ferreira; Diogo A. Gomes; Teruo Tada

doi:10.1090/proc/14475

Existence of weak solutions to first-order stationary mean-field games with dirichlet conditions

Rita Ferreira, Diogo A. Gomes, Teruo Tada

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

In this paper, we study first-order stationary monotone meanfield games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton-Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty's method, we show the existence of weak solutions to the original MFG.

Original language	English (US)
Pages (from-to)	4713-4731
Number of pages	19
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	11
DOIs	https://doi.org/10.1090/proc/14475
State	Published - Jul 24 2019

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): OSR-CRG2017-3452
Acknowledgements: The authors were partially supported by baseline and start-up funds from King Abdullah University of Science and Technology (KAUST) and by KAUST project OSR-CRG2017-3452.

Access to Document

10.1090/proc/14475

https://repository.kaust.edu.sa/bitstream/10754/627601/7/FGT_MFG_Dirichlet_final%20%281%29.pdf

Cite this

@article{e8f542beef374b05a5614c39d73da48b,

title = "Existence of weak solutions to first-order stationary mean-field games with dirichlet conditions",

abstract = "In this paper, we study first-order stationary monotone meanfield games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton-Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty's method, we show the existence of weak solutions to the original MFG.",

author = "Rita Ferreira and Gomes, {Diogo A.} and Teruo Tada",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): OSR-CRG2017-3452 Acknowledgements: The authors were partially supported by baseline and start-up funds from King Abdullah University of Science and Technology (KAUST) and by KAUST project OSR-CRG2017-3452.",

year = "2019",

month = jul,

day = "24",

doi = "10.1090/proc/14475",

language = "English (US)",

volume = "147",

pages = "4713--4731",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "11",

}

TY - JOUR

T1 - Existence of weak solutions to first-order stationary mean-field games with dirichlet conditions

AU - Ferreira, Rita

AU - Gomes, Diogo A.

AU - Tada, Teruo

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): OSR-CRG2017-3452 Acknowledgements: The authors were partially supported by baseline and start-up funds from King Abdullah University of Science and Technology (KAUST) and by KAUST project OSR-CRG2017-3452.

PY - 2019/7/24

Y1 - 2019/7/24

N2 - In this paper, we study first-order stationary monotone meanfield games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton-Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty's method, we show the existence of weak solutions to the original MFG.

AB - In this paper, we study first-order stationary monotone meanfield games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton-Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty's method, we show the existence of weak solutions to the original MFG.

UR - http://hdl.handle.net/10754/627601

UR - http://www.ams.org/proc/2019-147-11/S0002-9939-2019-14475-0/

UR - http://www.scopus.com/inward/record.url?scp=85073009501&partnerID=8YFLogxK

U2 - 10.1090/proc/14475

DO - 10.1090/proc/14475

M3 - Article

SN - 0002-9939

VL - 147

SP - 4713

EP - 4731

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 11

ER -

Existence of weak solutions to first-order stationary mean-field games with dirichlet conditions

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this