In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

The question of “correctness” of cardinal interpolation with shifted three-directional box splines is solved for arbitrary orders of the directional vectors. It is shown that the corresponding symbol can be viewed as a collection of curves with certain properties (convexity, increasing argument, etc.) which are investigated in detail. The method of proof involves an induction argument which is based on properties of the exponential Euler splines (studied in [6]).

We use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u =(u1, …, uN), . We also prove the boundedness of the minimiser of l1 by using the reverse Holder inequality.

A centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

In this paper we obtain some theorems of Bishop–Gromov type for tubes about a submanifold and semitubes about a real hypersurface of a Riemannian or Kaehler manifold with curvature bounded either from below or from above.

In this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.

Recent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.

In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating

with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables (x, y, δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

We investigate the singular problem

where Ω is a bounded smooth domain, k a bounded, nonnegative measurable function and v Ω 0. For the solution u to this problem, which is shown to exist if k(x) > 0 on some subset of Ω with positive measure, a uniform bound for |∇u| in Ω is derived when k(x) ≧ ψ (dist (x, ∂Ω)) with ψ (s)/sv ∈ Lp(0, a) for some a > 0, p > 1.

Let U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.

A version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.