We assume that the Stefan problem with underheating has a classical solution until the moment of contact of two distinct free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with underheating, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution that is the solution of the heat equation.

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, non-linear selection of the minimal speed occurs.

The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using two-dimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described by the GLSE.

We are interested in the steady-state Euler–Poisson system for a potential flow used in the mathematical modelling of plasmas and semiconductors. In the case of the quasi-neutral limit, boundary layers can appear. We study this limit by using an asymptotic expansion. We show the existence and uniqueness of each profile and give the justification of the asymptotic expansion up to any order.

We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity:

$$\Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2,$$

where $\nu>0$. In order to apply the MMP, it is crucial to obtain the asymptotic expansion of $u$ at $\infty$.

In this paper, we show that if $K(x)$ satisfies suitable conditions, then the quasilinear Neumann problem $-\Delta_pu+|u|^{p-2}u=K(x)|u|^{q-2}u$ in an exterior domain $\varOmega$ has at least two solutions, of which one is a positive ground-state solution and one is a nodal solution.

The steady-state diffusion problem is considered in a thin plate perforated periodically by many cylindrical holes of critical sizes. The plate is scaled to a plate of thickness 1. The asymptotic behaviour of the solution to the resulting rescaled equation is studied when the thickness of the original plate, the holes' size and period converge to 0. The phenomenon of dimension reduction occurs, i.e the limiting equation is posed in the cross-section only. The equation contains a linear term which describes the sink effect of the holes. This term depends on the relationship between the thickness, the period and the size of the perforations.

We consider the nonlinear elliptic eigenvalue problem $-\Delta u+u^p=\lambda u$ in $B_R$, $u>0$ in $B_R$, $u=0$ on $\partial B_R$, where $B_R:=\{|x|<R\}\subset\mathbb{R}^N$ ($N\ge2$) and $p>1$ is a constant. This equation is well known as a model equation of population density for some species when $p=2$. Here, $\lambda>0$ represents the reciprocal number of its diffusion rate and $\Vert u\Vert_1$ stands for the mass of the species. We establish the precise asymptotic formula for $\Vert u_\lambda\Vert_q$ as $\lambda\to\infty$, where $1\le q<\infty$. We also obtain the difference between $\Vert u_\lambda\Vert_\infty$ and $\Vert u_\lambda\Vert_q$ when $\lambda\gg1$.

We give the notion of a conjugate instant along a solution of the relativistic Lorentz force equation (LFE). Electromagnetic conjugate instants are defined as zeros of solutions of the linearized LFE with fixed value of the charge-to-mass ratio; equivalently, we show that electromagnetic conjugate points are the critical values of the corresponding electromagnetic exponential map. We prove a second-order variational principle relating every solution of the LFE to a canonical lightlike geodesic in a Kaluza–Klein manifold, whose metric is defined using the value of the charge-to-mass ratio. Electromagnetic conjugate instants correspond to conjugate points along the lightlike geodesic, and therefore they are isolated; based on such correspondence and on a recent result of bifurcation for light rays, we prove a bifurcation result for solutions of the LFE in the exact case.

The aim of this paper is to link the analytic results of Brezis et al., Demengel and Ignat relative to $W^{1,1}$-mappings from $B^n$ into $S^1$ to the measure-theoretical geometric results in our previous work. The paper also contains a few remarks about mappings in $W^{1,p}$, $p\geq2$, with values in $S^2$.

Given a regular two-dimensional local ring $(R,\mathfrak{m}_R)$ and a complete $\mathfrak{m}_{R}$-primary ideal $I\subset R$, we describe the (Zariski) factorization of the complete $\mathfrak{m}_{R}$-primary ideals $J\subset I$ of codimension $1$ in terms of the factorization of $I$.

In this paper, I prove that on every fractal structure with three vertices, for suitable weights, the renormalization operator has an eigenform.

Let $\tau_\varOmega$ denote the lifetime of Brownian motion in an open connected set $\varOmega\subset\mathbb{R}^m$. We obtain the asymptotic behaviour of the expected lifetime $\mathbb{E}_x^y[\tau_\varOmega]$ as $y\to x$, where the Brownian motion is conditioned to start at $x$ and to exit $\varOmega\setminus\{y\}$ at $\{y\}$.

This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem.

This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation.

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti–Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.

In this paper we obtain a Hermite-Hadamard type inequality for a class of subharmonic functions. Our proofs rely essentially on the properties of elliptic partial differential equations of second order. Our study extends some recent results from [1], [2] and [6].

Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of Kk? More specifically, let fk(n, m) be the largest integer t such that, for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies of Kk. Turán's theorem together with Wilson's Theorem assert that if . A conjecture of Erdős states that for all plausible m. For any ε > 0, this conjecture was open even if . Generally, f_k(n,m) may be significantly smaller than . Indeed, for k=7 it is easy to show that for m ≈ 0.3n2. Nevertheless, we prove the following result. For every k≥ 3 there exists γ>0 such that if then . In the special case k=3 we obtain the reasonable bound γ ≥ 10−4. In particular, the above conjecture of Erdős holds whenever G has fewer than 0.2501n2 edges.

A family of subsets of an n-set is 2-cancellative if, for every four-tuple {A, B, C, D} of its members, A∪ B∪C=A∪ B∪ D implies C = D. This generalizes the concept of cancellative set families, defined by the property that A∪B ≠A ∪ C for A, B, C all different. The asymptotics of the maximum size of cancellative families of subsets of an n-set is known (Tolhuizen [7]). We provide a new upper bound on the size of 2-cancellative families, improving the previous bound of 20.458n to 20.42n.

We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobás, Janson and Riordan, which enables us to give a new, and perhaps more revealing, proof of their result on a phase transition.

In this paper, we establish some inequalities among the Lp-centroid body, the Lp-polar projection body, the Lp-John ellipsoid and its dual, which are the strengthened version of known results. We also prove inequalities among the polar of the Lp-centroid body, the Lp-polar projection body, the Lp-John ellipsoid and its dual.