In this paper we study the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations, called the neutronic multigroup diffusion model, in a periodic heterogeneous domain. Such a model is used for studying the criticality of nuclear reactor cores. In a recent work in collaboration with Grégoire Allaire, it is proved that, under a symmetry assumption, the first eigenvector of the multigroup system in the periodicity cell controls the oscillatory behaviour of the solutions, whereas the global trend is asymptotically given by a homogenized diffusion eigenvalue problem. It is shown here that when this symmetry condition is not fulfilled, the asymptotic behaviour of the neutron flux, corresponding to the first eigenvector of the multigroup system, is dramatically different. This result enables to consider new types of geometrical configurations in practical nuclear reactor core computations.

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.

Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.

Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

We show that certain interpolation results for compact operators established by Cobos and co-workers cannot be extended to general closed operator ideals. We shall also characterize compactness of an embedding in terms of functions related to the classical K- and J-functionals of interpolation theory.

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

Let be the open unit disc in C. It is proved that a holomorphic embedding f : C2 can grow arbitrarily fast near b. It is also proved that a holomorphic embedding f : C C2 can grow arbitrarily fast near infinity.

We study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.

We consider the Dirichlet problem for the equation -Δpu = q(|x|)f(u) in an annulus Ω ⊂ Rn, n ≥ 1, where Δpu = div(|∇u|p−2∇u) is the p-Laplacian operator, p > 1. With no assumption on the behaviour of the nonlinearity f either at zero or at infinity, we prove existence and localization of positive radial solutions for this problem by applying Schauder's fixed-point theorem. Precisely, we show the existence of at least one such solution each time the graph of f passes through an appropriate tunnel. So, it is easy to exhibit multiple, or even infinitely many, positive solutions. Moreover, upper and lower bounds for the maximum value of the solution are obtained. Our results are easily extended to the exterior of a ball, when n > p.

In this article we present a proof of existence of a weak solution for the semigeostrophic system of equations, formulated as an active scalar transport equation with Monge-Ampère coupling, with initial data in . This is an extension of a 1998 result due to Benamou and Brenier, who proved existence with initial data in .

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second-order elliptic operators, these estimates hold true on bounded C1,1 domains. For higher-order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to non-cooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

We consider internal gravity waves in a stratified fluid layer with rigid horizontal boundaries and periodic boundary conditions on the sides at constant temperature with a small constant viscosity, modelled using the incompressible Navier-Stokes equations. Using operator-theoretic methods to study the damping rates of internal waves we prove there are non-oscillatory wave modes with arbitrarily small damping rates. We provide an asymptotic approximation for these non-oscillatory modes. Additionally, we find that the eigenvalues for damped oscillations are in an explicitly describable half-ring.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.

A recent paper of Ockendon et al. discusses the Fanno model for quasi-one-dimensional flow of gas in a tube, in situations where the flow is turbulent and the tube is long enough for wall drag to be important. Based on appropriate scalings and with associated boundary conditions they derive equations for similarity solutions and make predictions about travelling and evolving waves. In this paper the existence, uniqueness and asymptotic behaviour of these wave forms is proved rigorously. Techniques include shooting methods (both one- and two-parameter), appropriate changes of variables, and comparison techniques.

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

We study a class of BGK approximations of parabolic systems in one space dimension. We prove stability and existence of global solutions for this model. Moreover, under certain conditions, we prove a rigorous result of convergence toward the formal limit, by using compensated compactness techniques.

We consider the eigenvalue problem in an arbitrary OrliczSobolev space. We show that the existence of an eigenvalue can be derived from a generalized version of Lagrange multiplier rule. Our approach also applies to more general problems. We emphasize that no 2 condition is imposed.

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if (k, d) = (2, 0) and s < sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.