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 construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.

We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line.

The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

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.

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.

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.

This second part of the paper strengthens the descent theory described in the first part torational maps and arbitrary base fields. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the ‘dynamical Northcott’ Theorem 1.11 of Part I in sharp form.

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.

While the classification project for the simple groups of finite Morley rank is unlikely toproduce a classification of the simple groups of finite Morley rank, the enterprise has already arrived at a considerably closer approximation to that ideal goal than could have been realistically anticipated, with a mix of results of several flavors, some classificatory and others more structural, which can be combined when the stars are suitably aligned to produce results at a level of generality which, in parallel areas of group theory, would normally require either some additional geometric structure, or an explicit classification. And Bruno Poizat is generally awesome, though sometimes he goes too far.

The classes in Valiant's theory are classes of polynomials defined by arithmetic circuits. We characterize them by different notions of tensor calculus, in the vein of Damm, Holzer and McKenzie. This characterization underlines in particular the role played by properties of parallelization in these classes. We also give a first natural complete sequence for the class VPnb, the analogue of the class P in this context.

Dichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor–Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable case, and is a special case of a recent result of Tanovic.

We establish an identification result of the projective special linear group of dimension 2among a certain class of groups the Morley rank of which is finite.

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.

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.

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 examine the Black–Scholes partial differential equation for the pricing of a traded option (an American call option on an asset paying a continuous dividend) and make comparisons with other well known free boundary diffusion problems, such as the oxygen consumption problem. The pricing of American options can be viewed as a free boundary problem and is, therefore, inherently nonlinear. We consider the short and long time behaviour of the free boundary, present analytic results for the option value in such limits, and consider the formulation of the problem as a variational inequality, and its numerical solution.