91 research outputs found
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 +, 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
The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation
We consider the diffusive Hamilton-Jacobi equation with Dirichlet boundary conditions in two space dimensions, which
arises in the KPZ model of growing interfaces. For , 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 , 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 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 in the
normal direction and in the tangential direction .
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
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,
is a bounded and uniformly elliptic matrix and is convex in
and grows at most like , with 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 . This completes the recent results in \cite{GMP}, where the
existence of at least one solution in this class has been proved
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
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