Mathematical Models and Methods in Applied Sciences

The linear Reissner–Mindlin membrane-bending plate model is the rigourous Γ-limit for zero thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we use the natural nonlinear scaling for the displacements u and the linear scaling for the infinitesimal microrotations Ā ∈ 𝔰𝔬(3). We also provide formal calculations for other combinations of scalings by retrieving other plate models previously proposed in the literature by formal asymptotic methods as corresponding Γ-limits. No boundary conditions on the microrotations are prescribed.

In this paper, we study the homogenization of the system of partial differential equations describing the quasistatic shearing of heterogeneous thermoviscoplastic materials. We first present the existence and uniqueness of the solution of the above system. We then define "stable by homogenization" models as the models where the equations in both the heterogeneous problems and the homogenized one are of the same form. Finally we show that the model with non-oscillating strain-rate sensitivity which is submitted to steady boundary shearing and body force, is stable by homogenization. In this model, the homogenized (effective) coefficients depend on the initial conditions and on the boundary shearing and body force. Those theoretical results are illustrated by one numerical example.

Thermodynamical consistency of plasticity models is usually written in terms of the so-called "maximum dissipation principle". In this paper, we discuss constitutive relations for dissipative materials written through suitable generalized gradients of a (possibly non-convex) metric. This new framework allows us to generalize the classical results providing an interpretation of the yield function in terms of Hamilton–Jacobi equations theory.

This paper is devoted to the rigorous justification of the quasi-neutral limit of bipolar transient drift-diffusion models for semiconductors with p–n junctions in the multidimensional space. The general initial data and smooth sign-changing doping profiles with good boundary conditions are considered. The limit is performed rigorously by using multiple scaling asymptotic analysis, in which one main point is the construction of a more accurate approximate solution involving the effect of initial layer. The uniform estimates with respect to the scaled Debye length are obtained through the elaborate energy method and the relative entropy functional method.

We consider an eigenvalue problem for a certain type of quasi-linear second-order differential equation on the interval (0, ∞). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a self-focusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.

In this paper, we analyze a mathematical model focusing on key events of the cell invasion process. The three equations of the corresponding coupled system describe the behavior of the invasive cells, the extracellular matrix and the degradative enzymes. We employ a fix-point method and a priori estimates to prove local and global existence, uniqueness and regularity properties of the solutions. Our approach enable us to find estimates that are uniform in time. This is essential in proving, in the last part of the paper, new results that establish the asymptotic behavior of the solutions.