1,721,052 research outputs found
On the weak theory for mean field games systems
No full textMean field games theory describes the strategic interactions in a large population of similar agents whereas the strategy of the individuals depend on the distribution law of the state. The equilibria solve a system of partial differential equations where a backward Hamilton-Jacobi-Bellman equation for the value function is coupled with a forward Fokker-Planck equation for the mass distribution. If the cost criteria depend on the density of the distribution law, a theory of weak solutions is needed to handle mean field games systems, including new results concerning the Fokker-Planck equation with L-2 drift. Here we prove existence and uniqueness of weak solutions when the dynamics takes place in the whole space RN. We extend previous results obtained so far only for compact state space
A Note on the Sobolev and Gagliardo-Nirenberg inequality when p > N
No full textIt is known that the Sobolev space W-1,W-P(R-N) is embedded into LNP/(N-P)(R-N) if p < N and into L-infinity(R-N) if p > N. There is usually a discontinuity in the proof of those two different embeddings since, for p > N, the estimate parallel to u parallel to(infinity) <= C parallel to Du parallel to(N/P)(p)parallel to u parallel to(1-N/p)(p) is commonly obtained together with an estimate of the Holder norm. In this note, we give a proof of the L-infinity-embedding which only follows by an iteration of the Sobolev-Gagliardo-Nirenberg estimate parallel to u parallel to(N/(N-1)) <= C parallel to Du parallel to(1). This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes
On the turnpike property for mean field games
No full textWe consider the behavior of mean field games systems in the long horizon, under the assumption of monotonicity of the coupling term. Assuming that the Hamiltonian is globally Lipschitz and locally uniformly convex, we show that the time dependent solution is exponentially close to the ergodic stationary state in the long intermediate stages. This is evidence of the so-called exponential turnpike property for optimal control problems. Indeed, our proof follows a general approach which relies on the stabilization through the Riccati feedback of the associated linearized system
Traveling waves for a nonlocal KPP equation and mean-field game models of knowledge diffusion
Full text linkWe analyze a mean-field game model proposed by economists Lucas and Moll [J. Political Econ. 122 (2014)] to describe economic systems where production is based on knowledge growth and diffusion. This model reduces to a PDE system where a backward Hamilton–Jacobi–Bellman equation is coupled with a forward KPP-type equation with nonlocal reaction term. We study the existence of traveling waves for this mean-field game system, obtaining the existence of both critical and supercritical waves. In particular, we prove a conjecture raised by economists on the existence of a critical balanced growth path for the described economy, supposed to be the expected stable growth in the long run. We also provide nonexistence results which clarify the role of parameters in the economic model. In order to prove these results, we build fixed point arguments on the sets of critical waves for the forced speed problem arising from the coupling in the KPP-type equation. To this purpose, we provide a full characterization of the whole family of traveling waves for a new class of KPP-type equations with nonlocal and nonhomogeneous reaction terms. This latter analysis has independent interest since it shows new phenomena induced by the nonlocal effects and a different picture of critical waves, compared to the classical literature on Fisher–KPP equations
On the regularity of the total variation minimizers
No full textWe prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term f is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension N <= 7 and (in case of the global regularity) for convex domains
Absorption effects for some elliptic equations with singularities
No full textWe give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data, As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6)
Remarks on existence or loss of minima of infinite energy
No full textGiven a nonnegative bounded Radon measure mu on Omega C R-N, we discuss the existence or nonexistence of minima of infinite energy (so-called weak minima, T-minima, renormalized minima) for functionals like J(v) = integral(Omega) alpha(x, v)vertical bar Delta v vertical bar(p) dx - integral(Omega) v d mu where p > 1. In most of our results, alpha(x, s) is coercive. According to the behavior of s -> alpha(x, s) at infinity, existence or nonexistence of such minima is proved, and the convergence of approximating minima of regularized functionals is studied. Differences arise whether the measure charges or not sets of null p-capacity and/or alpha(x, s) blows-up at infinity. Lastly, some results are proved when alpha(x, s) degenerates at infinity
Uniqueness of solutions for some nonlinear Dirichlet problems
No full textWe consider here a class of nonlinear Dirichlet problems, in a bounded domain Omega of the form { -div(a(x, u)delu) + div(Phi(u)) = f in Omega, u = 0 on thetaOmega, investigating the problem of uniqueness of solutions. The functions Phi(s) and s --> a(x, s) satisfy rather general assumptions of locally Lipschitz continuity (with possibly exponential growth) and the datum f is in L-1(Omega). Uniqueness of solutions is proved both for coercive a(x, s) and for the case of a(x, s) degenerating for s large
Weak Solutions to Fokker–Planck Equations and Mean Field Games
No full textWe deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward- backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned
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