We answer some open problems on subhereditary radicals of associative rings. In particular, we show that the Brown–McCoy radical is not subhereditary and that every left or right subhereditary and left or right stable radical satisfies all these properties. The latter of these results concerns, in fact, the question whether some classes of reduced rings are closed under one-sided essential extensions. We characterize several such classes.

A nonlinear diffusion process modelling aggregative dispersal is combined with local (in space) population dynamics given by a logistic equation and the resulting growth-dispersal model is analysed. The nonlinear diffusion process models aggregation via a diffusion coefficient, which is decreasing with respect to the population density at low densities. This mechanism is similar to area-restricted search, but it is applied to conspecifics rather than prey. The analysis shows that in some cases the models predict a threshold effect similar to an Allee effect. That is, for some parameter ranges, the models predict a form of conditional persistence where small populations go extinct but large populations persist. This is somewhat surprising because logistic equations without diffusion or with non-aggregative diffusion predict either unconditional persistence or unconditional extinction. Furthermore, in the aggregative models, the minimum patch size needed to sustain an existing population at moderate to high densities may be smaller than the minimum patch size needed for invasibility by a small population. The tradeoff is that if a population is inhabiting a large patch whose size is reduced below the size needed to sustain any population, then the population on the patch can be expected to experience a sudden crash rather than a steady decline.

We present an extension of Besicovitch spaces, which has its origin in the works of Zhikov et al. and Kozlov and Oleinik. We show that these spaces have a similar behaviour to the Lp spaces, and we give a notion of derivative that allows us to treat partial differential problems in this frame. To the best of our knowledge, this is new, even for the usual Besicovitch spaces.

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.

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 .

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.

Degond and Markowich discussed the existence and uniqueness of a steady-state solution in the subsonic case for the one-dimensional hydrodynamic model of semiconductors. In the present paper, we reconsider the existence and uniqueness of a globally smooth subsonic steady-state solution, and prove its stability for small perturbation. The proof method we adopt in this paper is based on elementary energy estimates.

We study a boundary perturbation problem for a one-dimensional Schrödinger equation in which the potential has a regular singularity near the perturbed end point. We give the asymptotic behaviour of the eigenvalues under the perturbation. This problem arose out of the author's studies of singular elliptic operators in higher dimensions and we illustrate this point with an example. The class of potentials to which this method applies is larger than that covered by standard results, which assume uniform ellipticity of the operator or a perturbative term that is analytic in the perturbation parameter.

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H-measures defined on the space of positions and frequencies. The present paper is devoted to the investigation of the Tartar equation which describes the evolution of the H-measure μt associated with a sequence of oscillatory solutions of the linear transport equation in cases when a given solenoidal velocity field v(x, t) is sufficiently smooth. Here, (t, x, y) ∈ (0, T) × Ω × S1, 0 < T < +∞, Ω is a bounded open subset of R2 and S1 is the unit circle in R2, given coefficients Yij = Yij(y) are infinitely smooth.

Assuming that v belongs to , we establish the well posedness of Cauchy problem for the Tartar equation in the same measure class as the H-measures are in. For this purpose, we develop and use an extension of the theory of Lagrange coordinates for a case of non-smooth solenoidal velocity fields.

Let f : Mn×n → R ∪ {∞} be a function on the space Mn×n of n by n matrices that is invariant with respect to the left and right multiplication by proper orthogonal tensors. It is shown that f(A) ≤ f(Ā) if f is convex and the partial sums of the singular values of A, Ā ∈ Mn×n satisfy certain ordering inequalities. The same holds if f is rank 1 convex and the partial products of the singular values satisfy analogous inequalities. The proofs emphasize the roles of the ordered-forces inequalities and the Baker-Ericksen inequalities for invariant convex and rank 1 convex functions. As an application, the evaluation of the convex and lamination convex hulls of fully rotationally invariant sets by Dacorogna and Tanteri is simplified and similar results are given for sets invariant only with respect to the proper orthogonal group.

We consider minimization problems involving the Dirichlet integral under an arbitrary number of volume constraints on the level sets and a generalized boundary condition. More precisely, given a bounded open domain Ω ⊂ Rn with smooth boundary, we study the problem of minimizing ∫Ω |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = αi, i = 1,…, k, and a generalized boundary condition u ∈ K. Here, K is a closed convex subset of H1 such that ; the invariance of K under provides that the condition u ∈ K actually depends only on the trace of u along ∂Ω.

By a penalization approach, we prove the existence of minimizers and their Hölder continuity, generalizing previous results that are not applicable when a boundary condition is prescribed.

Finally, in the case of just two volume constraints, we investigate the Γ-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain Ω. It turns out that the resulting Γ-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy that concentrates along the jump, and a boundary integral term due to the constraint u ∈ K. In the particular case where K = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al.

The relaxation of certain time-evolution problems is investigated. As a conceptually simple example, we study elastically deformable bodies that undergo martensitic phase transformations. The movement of the phase boundaries is hindered by dry friction. The fundamental problem is that the phase distribution forms a highly oscillatory microstructure in space. Therefore, it is desirable to derive a coarse-grained system that describes the effective properties. We introduce a concept of relaxation of the evolution system and apply it to the case where only two phases occur and the elastic energy is quadratic. Finally, we present a candidate for the relaxation in the general case.

We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state a are governed by two asymptotic equations: the first-order asymptotic equation if a is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions. It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the ‘concentration-annihilation’ phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv2 for which v2 may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.