Bifurcation and segregation in quadratic two-populations mean field games systems

Cirant, Marco; Verzini, Gianmaria

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1145-1177.

We search for non-constant normalized solutions to the semilinear elliptic system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\nu \Delta v_i+g_i\left(v_j^2\right)v_i={\lambda }_i{v}_i,v_i>0\hfill & \text{in}\Omega \hfill \\ {\partial }_n{v}_i=0\hfill & \text{on}\partial \Omega \hfill \\ {\int }_{\Omega }{v}_i^2\mathrm{d}x=1,\hfill & 1\le i,j\le 2,j\ne i,\hfill \end{array}\right.\end{array}$$

where $\nu >0$, $\Omega \subset R^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $\lambda {}_i$ are unknown. This system is obtained, via the Hopf−Cole transformation, from a two-populations ergodic Mean Field Games system, which describes Nash equilibria in differential games with identical players. In these models, each population consists of a very large number of indistinguishable rational agents, aiming at minimizing some long-time average criterion. Firstly, we discuss existence of nontrivial solutions, using variational methods when $g_i\left(s\right)=s$, and bifurcation ones in the general case; secondly, for selected families of nontrivial solutions, we address the appearing of segregation in the vanishing viscosity limit, i.e.

$$\begin{array}{c}\hfill {\int }_{\Omega }{v}_1{v}_2\to 0\text{as}\nu \to 0.\end{array}$$

Classification : 35J47, 49N70, 35B25, 35B32
Mots clés : Singularly perturbed problems, normalized solutions to semilinear elliptic systems, multi-population differential games

Y. Achdou, M. Bardi and M. Cirant, Mean field games models of segregation. Preprint (2016). | arXiv | MR

A. Ambrosetti and G. Prodi,A primer of nonlinear analysis, vol. 34 of Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (1993). | MR | Zbl

T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems. Preprint (2015). | arXiv | MR

T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $R^3$. J. Math. Pures Appl. 106 (2016) 583–614. | DOI | MR | Zbl

H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation models: asymptotics and qualitative properties. Arch. Ration. Mech. Anal. 208 (2013) 163–200. | DOI | MR | Zbl

H. Berestycki, S. Terracini, K. Wang and J. Wei. On entire solutions of an elliptic system modeling phase separations. Adv. Math. 243 (2013) 102–126. | DOI | MR | Zbl

L. A. Caffarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Amer. Math. Soc. 21 (2008) 847–862. | DOI | MR | Zbl

L. A. Caffarelli and J.-M. Roquejoffre, Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames. Arch. Ration. Mech. Anal. 183 (2007) 457–487. | DOI | MR | Zbl

S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Physica D 196 (2004) 341–361. | DOI | MR | Zbl

M. Cirant, Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. 103 (2015) 1294–1315. | DOI | MR | Zbl

M. Conti, S. Terracini and G. Verzini, Nehari’s problem and competing species systems. Ann. Inst. Henri Poincaré Anal. Non Linéaire 19 (2002) 871–888. | DOI | Numdam | MR | Zbl

M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195 (2005) 524–560. | DOI | MR | Zbl

M. Conti, S. Terracini and G. Verzini, Uniqueness and least energy property for solutions to strongly competing systems. Interfaces Free Bound. 8 (2006) 437–446. | DOI | MR | Zbl

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971) 321–340. | DOI | MR | Zbl

E.N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species. J. Differ. Eq. 251 (2011) 2737–2769. | DOI | MR | Zbl

E. Feleqi, The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3 (2013) 523–536. | DOI | MR | Zbl

A. Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR | Zbl

M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–251. | DOI | MR | Zbl

M. Huang, P. E. Caines and R. P. Malhamé. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $ϵ$-Nash equilibria. IEEE Trans. Automat. Control 52 (2007) 1560–1571. | DOI | MR | Zbl

A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res. Part B: Methodol. 45 (2011) 1572–1589. | DOI

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 (2006) 619–625. | DOI | MR | Zbl

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343 (2006) 679–684. | DOI | MR | Zbl

J.-M. Lasry and P.-L. Lions, Mean field games. Jpn J. Math. 2 (2007) 229–260. | DOI | MR | Zbl

P.-L. Lions, Cours au collège de france. Available at http://www.college-de-france.fr.

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Comm. Pure Appl. Math. 63 (2010) 267–302. | DOI | MR | Zbl

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems. J. Eur. Math. Soc. (JEMS) 14 (2012) 1245–1273. | DOI | MR | Zbl

B. Noris, H. Tavares and G. Verzini, Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains. Anal. Partial Differ. Eq. 7 (2014) 1807–1838. | MR | Zbl

B. Noris, H. Tavares and G. Verzini, Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete Contin. Dyn. Syst. 35 (2015) 6085–6112. | DOI | MR | Zbl

V. Quitalo, A free boundary problem arising from segregation of populations with high competition. Arch. Ration. Mech. Anal. 210 (2013) 857–908. | DOI | MR | Zbl

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971) 487–513. | DOI | MR | Zbl

N. Soave and A. Zilio, Uniform bounds for strongly competing systems: the optimal Lipschitz case. Arch. Ration. Mech. Anal. 218 (2015) 647–697. | DOI | MR | Zbl

N. Soave, H. Tavares, S. Terracini and A. Zilio, Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping. Nonlinear Anal. 138 (2016) 388–427. | DOI | MR | Zbl

J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21 (2008) 305–317. | DOI | MR | Zbl