A quasilinear elliptic equation in ℝN of Hamilton-Jacobi-Bellman type is studied. An optimal criterion for uniqueness which involves only a lower bound on the functions is given. The unique solution in this class is identified as the value function of the associated stochastic control problem.

Several energies measuring jump discontinuities of a unit length gradient field are considered and are called defect energies. The main example is a total variation I(φ) of the hessian of a function φ in a domain. It is shown that the distance function is the unique minimiser of I(φ) among all non-negative Lipschitz solutions of the eikonal equation |grad φ| = 1 with zero boundary data, provided that the domain is a two-dimensional convex domain. An example shows that the distance function is not a minimiser of I if the domain is noncovex. This suggests that the selection mechanism by I is different from that in the theory of viscosity solutions in general. It is often conjectured that the minimiser of a defect energy is a distance function if the energy is formally obtained as a singular limit of some variational problem. Our result suggests that this conjecture is very subtle even if it is true.

Let denote a subring of the complex field that contains 1 and is closed under complex conjugation. It is shown that, with respect to the involution induced by word-reversal, the algebra over of a free monoid admits a trace and a separating family of star matrix representations. From the existence of a trace it is deduced that the aforementioned involution is special, in the sense of Easdown and Munn. Similar results hold for the algebra over of a free monoid with involution.

We provide an explicit example of a function that is homogeneous of degree one, rank-one convex, but not convex.

We classify multiplicative white noise perturbations k(·) dw, of generalised KPP equations and their effects on deterministic approximate travelling wave solutions by the behaviour of , the solutions of the stochastic generalised KPP equations converge to deterministic approximate travelling waves and if

being an associated potential energy, Фs a solution of the corresponding classical mechanical equations of Newton, D being a certain domain in R1 × Rr then the white noise perturbations essentially destroy the wave structure and force the solutions to die down.

For the case

(suppose the existence of the limit) we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations. Numerical solutions are included and give good agreement with theoretical results.

We propose here a way to extend to all the 1-set of ℝ2 the well-known affine length which was just defined for a C2 curve. Moreover, this leads us to define the affine dimension of a 1-set which can be used for discriminate rectifiable 1-sets from unrectifiable 1-sets of ℝ2.

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

In this paper we consider the pseudo-orbit tracing property for dynamical systems generated by iterations of random diffeomorphisms. We first define a type of hyperbolicity by means of a ‘random’ multiplicative ergodic theorem, and then prove our shadowing result by employing the graph transformation methods. That result applies to, for example, the case of small random diffeomorphisms type perturbations of hyperbolic sets of deterministic dynamical systems.

In this paper a one-parameter class of four-dimensional, reversible vector fields is investigated near an equilibrium. We call the parameter μ and place the equilibrium at 0. The differential at 0 is supposed to have ±iq, q > 0, as simple eigenvalues and 0 as a double, nonsemisimple eigenvalue. Our ultimate goal is to construct homoclinic connections of periodic orbits of arbitrary small size, in fact we shall show that the oscillations of the homoclinic orbits at infinity are bounded by a flat function of μ. This result receives its significance from the still unsolved question as to whether solutions exist which are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. First we construct the periodic solutions. In contrast to previous work, we find these in a full rectangle [0, K0] × ]0,μ0], where K measures the amplitude of the periodic orbits. Then we show that for each n ∈ ℕ there is a μn and a family of periodic solutions X(μ), μ ∈]0,μn[, of Size μn. To each of these solutions, we can find two homoclinic orbits, which are distinguished by their phase shift at infinity. One example of such a vector field occurs when describing the flow of an inviscid irrotational fluid layer under the influence of gravity and small surface tension (Bond number b < ⅓), for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalised solitary wave. Our work shows that there exist solitary waves with oscillations at infinity of order less than |μ|n for every n.

We explore some necessary conditions for quasiconvexity in an attempt to show that rank-one convexity does not imply quasiconvexity when the target space for deformations is two- dimensional. An interesting construction is presented, showing how rank-one directions may fit with each other, making the task harder than in higher dimensions.

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

It is shown that spectral properties of Sturm–Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.

This paper examines the Cauchy problem for a viscoelastic model with relaxation

with discontinuous, large initial data, where ½ ≦ μ <1, δ > 0 are constants. We first give a definition of admissible (or entropic) solutions to the system. Under this definition, we prove the existence, uniqueness and continuous dependence of the global admissible solution for the system. Our methods are essentially due to Kruzkov, and the requirement that f(u) is not badly degenerate (more precisely, meas {x: f″(x) = 0} = 0), needed previously when considering the global existence problem for the same system, is removed.