Results are obtained on the existence of positive solutions to the following elliptic system:

in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

I discuss implications of the rank-one convexity condition in a particular case.

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

Weighted Poincaré inequalities are established in any bounded domain D in ℝn (n ≧ 2), and their connection with the Minkowski content of ∂D is explored.

We show the existence of a timelike periodic trajectory for a time-dependent Lorentzian metric on ℝn × ℝ.

In this paper, using a recent generalisation of Morse Theory, we study the existence of periodic solutions of the Lagrangian equation (1.1) with subquadratic potential and asymptotically flat, nonconstant, time-dependent metric on ℝN. In Section 3, we get also an ‘alternative result’ about the minimal period or the existence of infinitely many solutions.

We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type:

and quasilinear heat equations:

The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations to N-dimensional quadratic operators are also considered.

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

If TRf(x) is the spherical partial sum of the Fourier transform of f and T*f(x) = SUPR > 0 | TRf(x)|, sufficient conditions are given on the non-negative weight function ω(x) which ensure that T* restricted to radial functionsis bounded on the Lorentz space Lp,s(Rn,ω) into Lp,q(Rn, ω) For power weights, these conditions are also necessary. The weight pairs (u,v) for which the generalised Stieltjes transform Sλ is bounded from LP,S(R+, v)into Lp,q(R+, u)are also characterised. These are an essential ingredient for the study of T*.

This paper is devoted to the study of the singular limit of the minimal solutions, as p → 1, of quasilinear Neumann problems involving p-Laplacian operators. It is established that the limit function is of bounded variation and is locally Höolder-continuous inside the domain.

Let φ:ℝ→ℝ be defined by φ(s) = |s|p−2s, with p > 1 a fixed number. We extend Sturm Comparison Theorem of the linear differential equation

to the nonlinear differential equation

by using the Wirtinger inequality. A Lyapunov inequality and some oscillation criteria of (E) are also given.

As a generalisation of the well-known result of Perron and Frobenius, it was shown by Rothblum [13] and independently by Richman and Schneider [12] that every nonzero matrix with non-negative entries has a basis of the root space corresponding to the maximal eigenvalue, represented by root vectors with non-negative entries. Krein and Rutman [9] showed that a positive compact nonquasinilpotent operator on a Banach lattice has a positive eigenvector corresponding to its spectral radius. As an extension of both results, we give sufficient conditions on such an operator in order that its spectral subspace corresponding to its spectral radius has a basis made exclusively of positive root vectors.

We establish lower and upper bounds which are valid for the overall conductivity of twodimensional composites. They are based on a method which modifies the so-called translation method in a way which makes it effectively much more flexible. When specialised to composites of n > 2 isotropic phases, the new bounds are often strictly better than all the previously known ones. From the mathematical point of view, the improvement is due mainly to a new regularity result in p.d.e.s [2]. From the physical point of view the latter can be interpreted as a result bounding in a suitable sense the fluctuations of the ‘electric field’.

Cardinal interpolation by integer translates of shifted three-directional box splines is studied. It is shown that, for arbitrary orders, k, l, m ∈ N of the directional vectors, this problem is correct if and only if the shift vector is taken from the hexagonal shift region (modulo translation with respect to the lattice Z2). This confirms a conjecture of S. D. Riemenschneider [9], and settles the problem studied in [5] for the special case k = l = m in full generality. The method of proof is from homotopy theory.

In a recent work, G. Anzellotti and the present authors introduced a notion of variation for functions defined over a rectifiable current. In this paper, we give a definition of a curvature varifold slightly different from that of Hutchinson (equivalent in the nonoriented case) and we study the variation properties, in the sense of [2], of the normal to a rectifiable current when the associated varifold is a curvature varifold.

In this paper, we provide sufficient conditions which guarantee the uniform stability as well as asymptotic stability of the positive equilibrium for a food limited population model with time delay.

Let f be an odd, C2 function on [− 1, 1], which vanishes at ± 1, and such that f′(O) < 0, f′ (±1) > 0 and u ↦ f(u)/u is increasing. Dang, Fife and Peletier [5] showed that there is a unique solution u with values in [−1, 1] of

which has the same sign as xy. The linearised operator around u is B defined by

It is proved here that the spectrum of B contains at least one negative eigenvalue, that all eigenfunctions corresponding to negative eigenvalues have the symmetries of the square, and that for Allen–Cahn's nonlinearity (f(u) = 2u3 − 2u), there is exactly one negative eigenvalue.