Nicoletta Tchou

Hamilton-Jacobi Equations on Networks as Limits of Singularly Perturbed Problems in Optimal Control: Dimension Reduction

We consider a family of star-shaped planar domains Ωϵ, made of N non intersecting semi-infinite strips of thickness ϵ and of a central region whose diameter is proportional to ϵ. As ϵ → 0, Ωϵ tends to a network 𝒢 made of half-lines sharing an endpoint O. We study infinite horizon optimal control problems in which the state is constrained to remain in . We prove that the value function tends to the solution of a Hamilton-Jacobi equation on 𝒢, with an effective transmission condition at O. The effective equation is linked to an optimal control problem.

Hamilton-Jacobi equations on networks

We consider continuous-state and continuous-time control problem where the admissible trajectories of the system are constrained to remain on a network. Under suitable assumptions, we prove that the value function is continuous. We define a notion of viscosity solution of Hamilton-Jacobi equations on the network for which we prove a comparison principle. The value function is thus the unique viscosity solution of the Hamilton-Jacobi equation on the network.

div-curl Type Theorem, H-Convergence, and Stokes Formula in the Heisenberg Group

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumins complex on contact manifolds.

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.

A Multiscale Numerical Method for Poisson Problems in Some Ramified Domains with a Fractal Boundary

We consider some elliptic boundary value problems in a self‐similar ramified domain of \mathbb{R}^2 with a fractal boundary with Laplace’s equation and nonhomogeneous Neumann boundary conditions. The goal is to approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For this, we propose a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum. A self‐similar finite element method is proposed and tested.

Trace Theorems for a Class of Ramified Domains with Self-Similar Fractal Boundaries

This work deals with trace theorems for a class of ramified bidimensional domains

A finite difference scheme on a non commutative group

\Omega

Boundary Value Problems in Ramified Domains with Fractal Boundaries

with a self-similar fractal boundary

Quasi-Linear Relaxed Dirichlet Problems

\Gamma^\infty