On a Bernoulli problem with geometric constraints

A Bernoulli free boundary problem with geometrical constraints is studied. The domain \Om is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints

Self-inductance coefficient for toroidal thin conductors

We consider the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter \eps>0. An explicit form of the singular part of the corresponding potential u\ue is given. This allows to construct the limit potential $u$ (as \eps\to 0) and an approximation of the inductance coefficient L\ue. We establish some estimates of the deviation u\ue-u and of the error of approximation of the inductance. The main result shows that L\ue behaves asymptotically as \ln\eps, when \eps\to 0

Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem

We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions.Comment: 9 page

Optimal Control for a Steady State Dead Oil Isotherm Problem

We study the optimal control of a steady-state dead oil isotherm problem. The problem is described by a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of mechanics of a continuous medium. Existence and regularity results of the optimal control are proved, as well as necessary optimality conditions.Comment: This is a preprint of a paper whose final and definitive form will appear in Control and Cybernetics. Paper submitted 24-Sept-2012; revised 21-March-2013; accepted for publication 17-April-2013. arXiv admin note: text overlap with arXiv:math/061237

A perturbation method for the numerical solution of the Bernoulli problem

International audienceWe consider the numerical solution of the free boundary Bernoulli problem by employing level set formulations. Using a perturbation technique, we derive a second order method that leads to a fast iteration solver. The iteration procedure is adapted in order to work in the case of topology changes. Various numerical experiments confirm the efficiency of the derived numerical method