Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games

Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to +$\infty$, have been recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions, they lead to a system of two coupled partial differential equations, a forward Bellman equation and a backward Fokker-Planck equations. Finite difference schemes for the approximation of such systems have been proposed in previous works. Here, we prove the convergence of these schemes towards a weak solution of the system of partial differential equations

Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations

We consider a class of stationary viscous Hamilton--Jacobi equations as \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. where \la\geq 0, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,\xi)$ is convex in $\xi$ and grows at most like $|\xi|^q+f(x)$, with $1 < q < 2$ and f \in \elle {\frac N{q'}}. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy--type estimate, i.e. (1+|u|)^{\bar q-1} u\in \acca, for a certain (optimal) exponent $\bar q$. This completes the recent results in \cite{GMP}, where the existence of at least one solution in this class has been proved

The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation

We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the precise final asymptotic profile, under the form $$u_y(x,y,T) \sim d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{as $(x,y)\to (0,0)$.}$$ Interestingly, this result displays a new phenomenon of strong anisotropy of the profile, quite different to what is observed in other blowup problems for nonlinear parabolic equations, with the exponents $1/(p-1)$ in the normal direction $y$ and $2/(p-2)$ in the tangential direction $x$. Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation, whereas the normal profile remains self-similar.Comment: Int. Math. Res. Not. IMRN, to appea

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation λu − div(A(x)Du) = H(x, Du) in Ω where A(x) is a bounded coercive matrix with measurable ingredients, λ ≥ 0 and ξ → H(x, ξ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H(x, ·) was required to hold globally