In recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the form

where Ω is an open, bounded subset of ℝN, q ≧1, g ∈ L∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the form

where the convex function h grows linearly at infinity.

In this paper, we consider the question of existence of solutions and their regularity properties for a large class of stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions together with noisy boundary data. They cover also stochastic boundary value problems. Our results are illustrated by two practical examples.

We consider the positive solutions to the semilinear equation:

where Ω denotes a smooth bounded region in ℝN(N > 1) and ℷ 0. Here f :[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such that f(0)<0 (semipositone). Assuming that f′(∞)≡limt→∞f(t)> 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] where f′(∞)≦0.

In this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

Positive definite temperature functions u(x, t) in ℝn+1 = {(x, t)| x ∈ ℝn,t > 0} are characterised by

where μ is a positive measure satisfying that for every ℰ > 0,

is finite. A transform is introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions in Then correspondence given by generalises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.

The special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

We consider the existence of positive solutions to a class of singular nonlinear boundary value problems for P-Laplacian-like equations. Our approach is based on the Schauder Fixed-Point Theorem.

In this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, where

a bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

We consider initial and boundary-value problems modelling the vibration of a plate with piezoelectric actuator. The usual models lead to the Bernoulli–Euler and Kirchhoff plate equations with right-hand side given by a distribution concentrated in an interior curve. We obtain regularity results which are stronger than those obtained by simply using the Sobolev regularity of the right-hand side. By duality, we obtain new trace regularity properties for the solutions of plate equations. Our results provide appropriate function spaces for the control of plates provided with piezoelectric actuators.

It is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

We analyse global existence of solutions to a system of two reaction–diffusion equations for whicha ‘balance’ law holds. The main aim is to make clear the influence of different combinations ofboundary conditions on global existence under the assumption that the nonlinearities satisfy polynomial growth estimates.

We show that the linear viscous damping Δut, is so strong that it altogether prevents propagation of singularities of the gradient of solutions to the system of viscoelasticity. Moreover, no creation or annihilation of singularities is possible in finite time.

We present a purely combinatorial procedure for finding an isolating neighbourhood and an index pair contained in a given set, being a finite union of cubes in Rs. It is applied for a computer-assisted computation of the Conley index of an isolated invariant subset of the Hénon attractor. As a corollary, it is shown that the Hénon attractor contains periodic orbits of all principal periods except for 3 and 5.

We give an existence theorem for a nonconvex minimum problem of the Calculus of Variations.

This paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.