H-convergence for equations depending on monotone operators in Carnot groups

On the Convergence of Maximal Monotone Operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch–Brezis type.

Homogenization and Convergence of Correctors in Carnot Groups

Homogenization of Hamilton-jacobi Equations in Carnot Groups

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups.The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property. 1. Background and motivations Consider a Hamilton-Jacobi equation: u+H(ξ,∇u) = 0 in R where the Hamiltonian H(ξ, p) : R × RN×N → R is not coercive in ...

Global convergence for ill-posed equations with monotone operators: the dynamical systems method

Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2 loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends t...

Convex Functions on Carnot Groups

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.