We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.

We study the self-similar solutions of the equation

in ℝN, when p > 2. We make a complete study of the existence and possible uniqueness of solutions of the form

of any sign, regular or singular at x = 0. Among them we find solutions with an expanding compact support or a shrinking hole (for t > 0), or a spreading compact support or a focusing hole (for t < 0). When t < 0, we show the existence of positive solutions oscillating around the particular solution .

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem

where Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.

There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.

The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).

We prove that the Lorentz sequence spaces do not have the property B of Lindenstrauss. In fact, for any admissible sequences w, v ∈ c0 \ l1, the set of norm-attaining operators from the Orlicz space hϕ(w) (ϕ is a certain Orlicz function) into d(v, 1) is not dense in the corresponding space of operators. We also characterize the spaces such that the subset of norm-attaining operators from the Marcinkiewicz sequence space into its dual is dense in the space of all bounded and linear operators between them.

We study the Benney equation and show that the associated initial-value problem is locally well-posed in Sobolev spaces Hs(ℝ2) for s > −2. Furthermore, we use a priori estimates to establish the global well-posedness for s ≥ 0. We also prove that these results are in some sense sharp. In addition, we obtain some exact travelling-wave solutions of the equation.

We prove that the triangle inequalities in general are not equivalent to lower semi-continuity of surface energy. In 1990, Ambrosio and Braides proved that the two conditions are equivalent when the number of regions, s, is three. They gave an example in the plane showing that the triangle inequalities do not imply lower semi-continuity when s ≥ 6. The cases when s = 4 and s = 5 have remained open. In this paper, we resolve these open questions and show that, in ℝm, the triangle inequalities are sufficient for lower semi-continuity if and only if s = 3.

We consider the diffusive logistic equation

supplemented by the nonlinear boundary condition

where α is a non-negative, non-decreasing function with α([0, 1]) ⊆ [0, 1]. When regarded as an ecological model for an organism inhabiting a focal patch of its habitat, the assumptions on α are intended to capture a tendency on the part of the organism to remain in the habitat patch when it encounters the patch boundary that increases with species density. Such a mechanism has been suggested in the ecological literature as a means by which the dynamics of the organism at the scale of the patch might differ from its local dynamics within the patch. Building upon earlier examinations of the boundary-value problem by Cantrell and Cosner, we detail in this paper the global disposition of biologically relevant equilibria when both 0 and 1 (the local carrying capacity within the patch) are equilibria. Our analysis relies on global bifurcation theory and estimates for elliptic and parabolic partial differential equations.

We consider a geometrically nonlinear model for crystal plasticity in two dimensions, with two active slip systems and rigid elasticity. We prove that the rank-1 convex envelope of the condensed energy density is obtained by infinite-order laminates, and express it explicitly via the 2F1 hypergeometric function. We also determine the polyconvex envelope, leading to upper and lower bounds on the quasiconvex envelope. The two bounds differ by less than 2%.

This paper is concerned with the existence and nonlinear stability of periodic travelling-wave solutions for a nonlinear Schrödinger-type system arising in nonlinear optics. We show the existence of smooth curves of periodic solutions depending on the dnoidal-type functions. We prove stability results by perturbations having the same minimal wavelength, and instability behaviour by perturbations of two or more times the minimal period. We also establish global well posedness for our system by using Bourgain's approach.

We prove sharp upper bounds for invariant measures of Markov processes in ℝN associated with second-order elliptic differential operators with unbounded coefficients.

The main goal of this paper is to obtain the lower bounds

for solutions uν of nonlinear Schrödinger equations with small viscosity ν. We also discuss the application of the above estimate to the theory of turbulence. Namely, we are interested in time-averaged lower bounds, which are important in establishing upper bounds for the turbulent space-scale.

We study isometries between weighted spaces of holomorphic functions on unbounded domains in ℂn. We show that weighted spaces of holomorphic functions on unbounded domains may exhibit behaviour different from that observed on bounded domains. We calculate the isometries for specific weights on the complex plane and the right half-plane.

In this article, we study the existence of travelling waves for a class of epidemic models structured in space and with respect to the age of infection. We obtain a necessary and sufficient condition for the existence of travelling waves for such a class of problems. As a consequence of our main result, we also derive the existence of travelling waves of a class of functional partial derivative equations.

We prove that for all x > 0, we have

with the best possible constants α = 0 and .

We study the normality of families of meromorphic functions defined in terms of certain omitted functions. In particular, we prove the following results. Firstly, if is a family of meromorphic functions in a domain D ⊂ ℂ, and a(z), b(z) and c(z) are distinct meromorphic functions in D and if, for all f ∈ and all z ∈ D, f(z) ≠ a(z), f(z) ≠ b(z) and f(z) ≠ c(z), then is normal in D. Secondly, letting R(w) be a rational function of degree greater than or equal to 3 and be a family of functions meromorphic in a domain D ⊂ ℂ, if there exists a non-constant meromorphic function α(z) in D such that, for all f ∈ and z ∈ D, R(f(z)) ≠ α(z), then is normal in D.

We use spectral sequence techniques to compute centralizers of elements within graded Lie algebras, and the methods are then applied to the calculation of unique normal forms of elements within one-parameter matrix Lie algebras. A finiteness criterion for unique normal forms is presented.

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

A variational problem for a functional with slowly growing principal part and involving critical Orlicz–Sobolev lower term with respect to the principal part is discussed. The principal part of the functional is not Fréchet differentiable. The lack of differentiability and the critical growth rate of the lower term demand a precise compactness argument in the variational approach. A non-negative solution for the Euler equation is given.

We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.