We define and study families of conjugate and reflected curve congruences associated to a self-adjoint operator on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the -asymptotic and the -principal curves, and the pencil joining the -characteristic and the -principal curves, respectively. The binary differential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE, of its -conjugate equation and on BDEs with orthogonal roots.

We study a generalization of the Svarc genus of a fibre map. For an arbitrary collection ɛ of spaces and a map f : X → Y, we define a numerical invariant, the ɛ-sectional category of f, in terms of open covers of Y. We obtain several basic properties of ɛ-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple axioms which characterize the ɛ-sectional category. In the final section, we obtain inequalities for the ɛ-sectional category of a composition and inequalities relating the ɛ-sectional category to the Fadell–Husseini category of a map and the Clapp–Puppe category of a map.

We consider the equation

where

Under these conditions, (1) is correctly solvable in L1(ℝ), i.e.

(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);

(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),

In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimate

where c2 is some absolutely positive constant.

This note is concerned with the spaces F'p,µ of generalised functions introduced in a previous paper. A necessary and sufficient condition for an inclusion of the form

to hold is established. The case p = ∞ leads to consideration of a class G''∞µ whose simple properties are noted. Some consequences of relevance to fractional integrals and Hankel transforms are indicated.

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.

Calculations are made of the resonance energy, bond order and bond length in a series of graphitic layers of varying size. Carbon–carbon bond lengths appear to vary very little in size with increasing number of carbon atoms, in agreement with experiment. But variations in resonance energy are significant, and indicate clearly that resonance, by itself, favours an approximately square, rather than oblong, shape. But in the case of such layers in equilibrium in the presence of molecular hydrogen, the most stable layer containing a given number of carbon atoms is of the long, thin polyphenyl type. Some tentative calculations suggest that polymerisation of smaller groups to larger ones should be endothermic, in agreement with the experimental fact that the formation of larger graphitic crystallites during carbonisation occurs, with emission of hydrogen, only at high temperatures.

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

We prove the stability with respect to the flux of solutions to initial – boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hörmander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1, ∞).

We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.

A class of models of single-species dynamics with diffusion within and between patches is considered. It is shown that under our prescribed conditions, there is a unique, positive, globally asymptotically stable steady state.

We show the incompressible Navier–Stokes–Maxwell system with solenoidal Ohm's law can be derived from the two-fluid incompressible Navier–Stokes–Maxwell system when the momentum transfer coefficient tends to zero. The strategy is based on the decay and dissipative properties of the electromagnetic field.

A method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.

Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.

In this paper, we give a positive answer to this question.

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.

Gramsch and Lay [8] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$,

\[ \widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), \]

$\widetilde {\sigma }_i$

$f$

$f$

$\widetilde \sigma$

In this note, we prove two monotonicity formulas for solutions of $\Delta _H f = c$ and $\Delta _H f - \partial _t f = c$ in Carnot groups. Such formulas involve the right-invariant carré du champ of a function and they are false for the left-invariant one. The main results, theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt–Caffarelli–Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an ‘almost monotonicity’ formula be possible. In the last section, we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.

Page 277, line 14: Read “measure” instead of “distribution”.

Page 287, line 1: Read “(Ω)” instead of “′(Ω)”,

line 13: Read “explanation” instead of “generalization”,

line 23: Read “In fact, Corollary 1” instead of “In fact, Corollary 2”.

We thank Professor L. Véron for directing our attention to these mistakes.