In this paper, we prove the existence of at least one solution to the problem

where ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

A boundary integral method is developed for the scattering of electromagnetic waves at thin obstacles. The exterior boundary value problem for the vector Helmholtz equation with given Neumann data on an open surface piece (screen S) is converted into a system of integral equations for the jumps of the tangential component of the field and its divergence across the screen. A slight modification of the Cauchy data yields a strongly elliptic system of pseudodifferential equations on S which can therefore be used for numerical computations using Galerkin's procedure. The resulting boundary integral equations are analysed using pseudodifferential operator calculus. The principal symbol concept, together with the Wiener–Hopf technique, are used to derive existence and regularity results for the solutions to the boundary integral equations. Quasi-optimal error estimates in the energy norm are given for the numerical scheme.

In this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.

We show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

Initial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.

We consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion. Based on the results in another paper by the same author, a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation (the Semenov approximation).

We derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equation

in the case where the only restriction on the real-valued function q is

In this paper, we study the converse of comparison results for solutions to linear second-order elliptic equations. Namely, in the inequalities proved by G. Talenti and others, we study the case of equality and prove that “equalities are achieved only in the spherical situation”. We also present some applications of these results to semilinear elliptic equations.

In this paper we establish conditions to prove that if classical solutions to the initial boundary value problems for nonlinear elastodynamics exist, then they depend Hölder continuously on their initialdata and body forces.

In an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equation

has the asymptotic expansion

as |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.

We show that if the real valued function q admits the expansion

in a neighbourhood of 0, then

A Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

Let Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

Two alternative characterizations of semidirect products of semigroups are given. Characterizations are provided of such products that are groups, regular semigroups, and inverse semigroups, respectively.

This paper presents some regularity results on the solution and on the free boundary for the one phase Stefan problem with zero specific heat in the framework of the variational inequalities formulation. In particular we show the Hölder continuity of the free boundary. Estimates on the rate of convergence when the specific heat vanishes are given for the variational solutions and for the free boundaries.

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.

It is proved that if a ring R has the extension property in containing ringSi, then the amalgam [R; Si,] is strongly embeddable. Using a result of P. M. Cohn, a necessary and sufficient condition for a ring to be an amalgamation base is then given. It is also shown that R is a level subring of a ring S if and only if S/R is flat. From this, a classical result of P. M. Cohn on flat amalgams is proved.

We demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

Let B(k) be a linear bounded mapping of a Banach space X into a Banach space Y meromorphic in the parameter k on a connected domain of the complex plane. Under certain assumptions on B(k), more general than previously considered, the singularities of the inverse operator are described.

We give generalizations of the Landau–Hadamard inequality ‖u′‖2 ≦ K ‖u‖ ‖u″‖ replacing u” by the second-order differential expression u″ − (α + β)u′ + αβu (α, β ∈ ℂ). The new functional inequalities are then used to obtain similar inequalities for dissipative and skew-Hermitian operators.

We consider a perturbed version of a membrane problem when the latter has no solution. The modified problem has a solution which depends on a parameter ε > 0. As ε →0 this solution is seen to tend towards a “generalised solution” of the original problem.