Bento Louro

Asymptotic analysis of reaction-diffusion-electromigration systems

We consider a Nernst-Planck-Poisson system modelling ion migration through biological membranes, in the one dimensional case. The model includes both the effects of biochemical reaction between ions and of fixed charges. We state the existence of solutions under either an imposed potential condition or an imposed current condition. We study the asymptotical behaviour of solutions in the limit of electroneutrality. Non uniform convergence gives rise to jump in the potential, known as Donnan potential. Finally we give correctors which describe the charged boundary layers.

Factorization of Overdetermined Boundary Value Problems

The purpose of this chapter is to present the application of the factorization method of linearelliptic boundary value problems to overdetermined problems. The factorization method ofboundary value problems is inspired from the computation of the optimal feedback control inlinear quadratic optimal control problems. This computation uses the invariant embeddingtechnique of R. Bellman (1): the initial problem is embedded in a family of similar problemsstarting from the current time with the current position. This allows to express the optimalcontrolasalinearfunctionofthecurrentstatethroughagainthatisbuiltusingthesolutionofa Riccati equation. The idea of boundary value problem factorization is similar with a space-wise invariant embedding. The method has been presented and justified in (5) in the simplesituation of a Poisson equation in a cylinder. In this case the family of spatial subdomains issimply a family of subcylinders. The method can be generalized to other elliptic operatorsthan the laplacian (7) and more general spatial embeddings (6). The output of the method isto furnish an equivalent formulation of the boundary value problem as the product of twouncoupled Cauchy initial value problems that are to be solved successively in a spatial di-rection in opposite ways. These problems need the knowledge of a family of operators thatsatisfy a Riccati equation and that relate on the boundaries of the subdomains the Dirichletand Neumann boundary conditions. This factorization can be viewed as an infinite dimen-sionalgeneralizationoftheblockGauss

Factorization by Invariant Embedding of a Boundary Value Problem for the Laplace Operator

This work concerns the factorization of a second order elliptic boundary value problem defined in a star-shaped bounded regular domain, in a system of uncoupled first order initial value problems, using the technique of invariant embedding. The family of domains is defined by a homothety. The method yields an equivalent formulation to the initial boundary value problem by a system of two uncoupled Cauchy problems. The singularity at the origin of the homothety is studied.

Factorization by Invariant Embedding of Elliptic Problems in a Circular Domain

We present a method to factorize a second order elliptic boundary value problem in a circular domain, in a system of uncoupled first order initial value problems. We use a space invariant embedding technique along the radius of the circle, in both an increasing and a decreasing way. This technique is inspired in the temporal invariant embedding used by J.-L. Lions for the control of parabolic systems. The singularity at the origin for the initial value problems is studied.

ON A NEW SCALING FOR SEMICONDUCTOR DEVICE EQUATIONS AND ITS ASYMPTOTIC ANALYSIS

The aim of this paper is to present a new scaling which is appropriate for modeling reverse biased semiconductor devices with moderately high applied potential and its asymptotic analysis. This scaling was motivated by the modeling of oxygen sensors. It is compared to the ones leading to (a) electroneutrality, (b) the existence of a depletion zone. With our scaling such a zone may appear within a boundary layer. A particular attention is paid to the qualitative dependence of the asymptotic solution to the boundary value of the concentration of carrier.

On the analysis of an endothermal/exothermal saturation problem

We consider an endo-or exo-thermal saturation problem that corresponds to a parabolic quasi-variational inequality. Applying regularity results and inequalities of Lewy-Stampacchia type, we prove the solvability of a modified problem (including the Steklov averaging and the mollification of the saturation velocity) for the nonlinear case and also of the exact problem for the linear case with a small coefficient in the temperature equation. Bibliography: 8 titles.