Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is driven only by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth-order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a 1½-dimensional situation.

We consider the stationary equations of a general viscous fluid in an infinite (periodic) slab supplemented with Navier's boundary condition with a friction term on the upper part of the boundary. In addition, we assume that the upper part of the boundary is described by a graph of a function φε, where φε oscillates in a specific direction with amplitude proportional to ε. We identify the limit problem when ε → 0, in particular, the effective boundary conditions.

For m > 0 and p ∈ (1, (N + 2)/(N − 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem:

In this work we describe the isoperimetric regions in complete symmetric annuli of revolution with Gauss curvature non-decreasing from the shortest parallel. This description allows us to complete the classification of isoperimetric regions in quadrics of revolution.

We study the effect of the potential |y|α on the stability of entire solutions for elliptic equations on ℝN, N ≥ 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a by-product of our results, we will improve the known results about the regularity of such solutions.

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary-value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev.

Consider a space-like plane Π in Minkowski space. Under the presence of a uniform time-like potential directed towards Π, this paper analyses the configurations of shapes that show a space-like surface supported in Π with prescribed volume and show that it is a critical point of the energy of this system. Such a surface is called stationary and it is determined by the condition that the mean curvature is a linear function of the distance from Π and the fact that the angle of contact with the plate Π is constant. We prove that the surface must be rotational symmetric with respect to an axis orthogonal to Π. Next, we show existence and uniqueness of symmetric solutions for a prescribed angle of contact with Π. Finally, we study the shapes that a stationary surface can adopt in terms of its size. We thus derive estimates of its height and the enclosed volume by surface with the support plane.

We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω ⊂ ℝn → ℝn is discrete and open, provided that its distortion function if n = 2 and that for some p > n − 1 if n ≥ 3.

This paper is concerned with the existence of a global attractor for a semi-flowgoverned by the weak solutions to a nonlinear one-dimensional thermoviscoelasticsystem with clamped boundary conditions in shape memory alloys. The constitutiveassumptions for the Helmholtz free energy include the model for the study ofmartensitic phase transitions in shape memory alloys. To describe physicallyphase transitions between different configurations of crystal lattices, we workin a framework in which the strain u belongsto L∞. New approachesare introduced and more delicate estimates are derived to establish the crucialL∞ -estimate ofstrain u in the course of showing thecompactness of the orbit of the semi-flow and existence of an absorbing set.

A one-dimensional Ginzburg–Landau model that describes a superconducting closed thin wire with an arbitrary cross-section subject to a large applied magnetic field is derived from the three-dimensional Ginzburg–Landau energy in the spirit of Γ-convergence. Our result proves the validity of the formal result of Richardson and Rubinstein, which reveals the double limit of a large field and a thin domain. An additional magnetic potential related to the applied field is found in the limiting functional, which yields a parabolic background for the oscillatory phase transition curve between the normal and superconducting states.

This paper concerns the asymptotic behaviours of pulse-like solutions for a 3 × 3 semilinear hyperbolic system in the limit of short wavelength ε. When two pulses interact with each other, we construct a pulse-like approximate solution up to Ο(ε), at which order a new pulse appears. The existence of a solution to the 3 × 3 semilinear problem with the initial data being the interaction of two pulses in a domain independent of the wavelength is proved in the space of co-normal distributions. Meanwhile, we obtain that the error between this exact solution and the approximate solution is of Ο(ε2) as ε → 0, which rigorously shows that there are three pulses propagated after the interaction of two pulses for the 3 × 3 semilinear system.