In this paper we study the asymptotic behaviour of the solution for a nonconvex relaxation model. The time decay rates in both the exponential and algebraic forms of the travelling wave solutions are shown by the weighted energy method. Our results develop and improve the stability theory in [8,9].

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

Let X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p > 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.

We study fine properties of currents in the framework of geometric measure theory on metric spaces developed by Ambrosio and Kirchheim, and we prove a rectifiability criterion for flat currents of finite mass. We apply these tools to study the structure of the distributional Jacobians of functions in the space BnV, defined by Jerrard and Soner. We define the subspace of special functions of bounded higher variation and we prove a closure theorem.

We provide new sufficient conditions for the existence of multiple fixed points for a map between ordered Banach spaces. An interesting feature of this approach is that we require conditions not on two boundaries, but rather on one boundary and a point with some extra information on the monotonicity of the nonlinearity on a certain set. We apply our results to prove the existence of at least two positive solutions for a nonlinear boundary-value problem that models a thermostat.

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

Uniqueness of non-negative solutions conjectured in an earlier paper by Shivaji is proved. Our methods are independent of those of that paper, where the problem was considered only in a ball. Further, our results apply to a wider class of nonlinearities.

We study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.

Let A be a unital C*-algebra and let B be an abelian C*-subalgebra containing the identity of A. For any pure state h of B let Fh be the set of states of A which restrict to h on B. Necessary and sufficient conditions are given for an element x in A to have the property that, for each h, x is unable to distinguish between distinct elements of Fh. By specializing, this leads to a new proof of a theorem giving necessary and sufficient conditions for Fh to be a singleton for each h.

It is also shown that if A is postliminal and π(B) is a maximal abelian C*-subalgebra of π(B) for each irreducible representation π of A then Fh is a Choquet simplex for each h.

During the present session I propose to lay before the Society several papers on subjects connected with the deposits which were found at the bottom of the oceans and seas visited by H.M.S.Challenger in the years 1872, 1873, 1874, 1875, and 1876.

In a paper recently communicated to the London Mathematical Society the present author showed that the characteristic numbers an and bn of the Mathieu equation—

may be developed, for large positive values of q, in the forms

A physically transparent and mathematically streamlined derivation is presented for a third-order nonlinear dynamical system that describes the curious chiral reversals of a celt (rattleback). The system is integrable, and its solutions are periodic, showing an infinite succession of spin reversals. Inclusion of linear dissipation allows any given number of reversals, and a typical celt's observed behaviour is well captured by tuning the dissipation parameters.

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 study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.

It is well known that if f, g, h are nonnegative functions and f*, g*, h* their symmetrically decreasing rearrangements, then

also if u* is a spherically decreasing rearrangement of a function u,

In this paper it is proved under suitable assumptions (including the assumption that h is already rearranged) that equality holds in (i) if and only if f and g are already rearranged, and, if 1 < p < ∞ equality holds in (ii) if and only if u is already rearranged. We discuss (ii) both in ℝn and on the unit sphere.

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

The electrical heating of a conductor connected with a current limiting device of total resistance R is studied under mixed boundary conditions. A theorem of existence is given using a transformation which permits the reduction of the nonlinear system governing the problem to the classical mixed boundary value problem for the Laplace equation.

Aus Leptenychotes weddelli Lesson. Ventric. Das Kopfende ist erheblich diinner als das Schwanzende. Die Lippen sind klein, die Dorsallippe ist 0·18 mm. breit und 0·14 mm. lang; aussen an der Vorderseite gerundet, Papillen nach aussen und vorn gerückt, innen mit einem doppelten, rechtwinklig begrenzten Vorsprung, ohne Zahnleisten, Löffelbildung und Zwischenlippen (fig. 1). Cuticula quergeringelt, Nacken - papillen fehlen; Schwanzende abgerundet, beim Weibchen sehr breit. Der Ösophagus ist kurz und schmal und nimmt der Gesammtlange länge ein; in seiner vorderen Hälfte ist er muskulös, in der hinteren drüsig, das Lumen ist aber überall drei-schenklig; immuskulösen Theil verlaufen 3 Drüsen (fig. 3, a); an der Dorsalseite der Ösophagus verläuft eine breite, blinddarmartige Vörlangerung des Darms nach vorn, die weit nach vorn reicht (fig. 3, b); nieht so lang wie der ösophagus sondern von der Ösophagus-Länge; die das Lumen auskleidende Membran sowie die äussere Hülle sind dick.

The Kuramoto–Sivashinsky-perturbed Korteweg–de Vries (KS–KdV) equation

$$ \partial_tu=-\partial_x^3u-\tfrac{1}{2}\partial_x(u^2)-\varepsilon(\partial_x^2+\partial_x^4)u, $$

with $0<\varepsilon\ll1$ a small parameter, arises as an amplitude equation for small amplitude long waves on the surface of a viscous liquid running down an inclined plane in certain regimes when the trivial solution, the so-called Nusselt solution, is sideband unstable. Although individual pulses are unstable due to the long-wave instability of the flat surface, the dynamics of KS–KdV is dominated by travelling pulse trains of $O(1)$ amplitude. As a step toward explaining the persistence of pulses and understanding their interactions, we prove that for $n=1$ and $2$ the KdV manifolds of $n$-solitons are stable in KS–KdV on an $O(1/\varepsilon)$ time-scale with respect to $O(1)$ perturbations in $H^n(\mathbb{R})$.

The exterior Dirichlet problem for the Helmholtz equation in two dimensions is reduced to a boundary integral equation which is soluble by iteration. A standard application of Green's theorem leads to boundary integral equations which are not uniquely soluble because the operator has an eigenvalue. The present approach modifies the operator in such a way that the former eigenvalue is in the resolvent spectrum for low frequencies. The results are applied to the inverse scattering problem wherein the far field is known for a limited frequency range and one seeks the curve on which a plane wave is incident and a Dirichlet boundary condition is assumed. The first iterate in the solution of the boundary integral equation is used to obtain a sequence of moment problems relating the Fourier coefficients of the far field to the coefficients of the Laurent expansion of the conformai transformation which maps the exterior of a circle onto the exterior of the unknown curve. These moment problems are soluble in terms of the mapping radius which in turn may be determined from scattered far field data for an incident plane wave from a second direction.