The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.

However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.

This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).

We consider variational problems of the form where Ω is a bounded open set in RN, f : RN → R is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum u0 is any function in W1, p (Ω). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove the existence of solutions to ( P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R.

In this work we study the existence and qualitative properties of non-negative solutions of the Lotka—Volterra models with nonlinear diffusion under homogeneous Dirichlet boundary conditions. We consider the three typical interactions: prey—predator, competition and symbiosis. Unlike the linear diffusion models, non-trivial non-negative solutions can exist which are not strictly positive. Sufficient conditions in terms of the coefficients involved in the setting of the models are given, assuring that one species (or both) does not survive on a set of its habitat (called ‘dead core’) of positive measure.

The paper is concerned with nonlinear equations of critical Sobolev growth involving the p-Laplace operator. These equations generalize the more classical scalar curvature equation.

This note develops a connection formula for spectral functions given in a recent paper by Gilbert and Harris.

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we are able to prove existence and uniqueness results for solutions of nonlinear RiemannHilbert problems with non-compact Lipschitz continuous restriction curves.

This paper is concerned with the existence of maximizers for a certain non-convex energy functional, relative to a class of rearrangements of a given function. Physically, the solution represents a uniform shear flow containing a bounded vortex anomaly, in R2. The prescribed data are the rearrangement class of the vorticity field.

This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions is said to be refinable if it satisfies the vector refinement equation where a is a finitely supported sequence of r × r matrices called the refinement mask. A complete characterization for the local linear independence of the shifts of ϕ1,…,ϕr is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets.

The initial-boundary value problem for the non-homogeneous Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

We study the behaviour, as t → ∞, of solutions to the convectiondiffusion equation on the half-line with the homogeneous Neumann boundary condition and with bounded initial data. The higher-order terms of the asymptotic expansion in Lp (R+) of solutions are derived.

We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko's model may vanish.

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

Consider an open bounded connected set Ω in Rn and a Lebesgue measurable set E ⊂⊂ Ω of positive measure. Let u be a solution of the strictly elliptic equation Di (aij Dj u) = 0 in Ω, where aij ∈ C0, 1 (Ω̄) and {aij} is a symmetric matrix. Assume that |u| ≤ ε in E. We quantify the propagation of smallness of u in Ω.