Mathematical Models and Methods in Applied Sciences

Let Ω be an open bounded set of ℝ³ and let W and V be two non-negative continuous functions vanishing at α, β and α', β', respectively. We analyze the asymptotic behavior as ε → 0, in terms of Γ-convergence, of the following functional:

We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, and quasi-periodic solutions.

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω₀. This question has been studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω₀. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

We study a new class of electromagnetostatic problems in the variational framework of the subspace of W^1,p(Ω) of vector functions with zero divergence and zero normal trace, for , in smooth, bounded and simply connected domains Ω of ℝ³. We prove a Poincaré–Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors.

We present a Cosserat-based three-dimensional to one-dimensional reduction in the case of a thin rod, of the viscoelastic finite strain model introduced by Neff. This model is a coupled minimization/evolution problem. We prove the existence and uniqueness of the solution of the reduced minimization problem. We also show a few regularity results for this solution which allow us to establish the well-posedness of the evolution problem. Finally, the reduced model preserves observer invariance.

We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good.

Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.