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.

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 $I$.

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 $S^2$.

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 $\infty$.

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.

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.

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.

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

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 $\lambda\gg1$.

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 $\{y\}$.

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.