We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

We show the $L^1$ contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem $j(v)_t-\text{div}(\nabla w+F(w))=f(t,x)$, $w=\varphi(v)$ in $(0,T)\times\varOmega\subset \mathbb{R}^+\times\mathbb{R}^N$ with inhomogeneous Dirichlet boundary datum $g\in L^2(0,T;W^{1,2}(\varOmega))$ for $w$ (the boundary datum is taken in the sense $w-g\in L^2(0,T;H^{1}_0(\varOmega))$) and initial datum $j_o\in L^1(\varOmega)$ for $j(v)$. Here $\varphi$ and $j$ are non-decreasing, and we assume that $F$ is just continuous.

Our proof consists in doubling of variables in the interior of $\varOmega$ as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of $\varOmega$. For the latter argument, the result is restricted to the linear dependence on $\nabla w$ of the diffusion term. The proof allows for a wide class of domains $\varOmega$, including, for example, weakly Lipschitz domains with Lipschitz cracks.

We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators.

In this paper we study some very weak notions of differentiability arising in connection with the spatial regularity of flows associated with non-smooth vector fields. The main difference from other similar concepts, also studied in a Sobolev setting, is that the convergence of difference quotients has to be understood as convergence in measure. We show in particular that the classical approximate differentiability is a property strictly stronger than approximate differentiability in measure.

The main theorem states that a bounded linear operator $h$ from a unital $C^{\ast}$-algebra $A$ into a unital Banach algebra $B$ must be a homomorphism provided that $h(\bm{1})=\bm{1}$ and the following condition holds: if $x,y,z\in A$ are such that $xy=yz=0$, then $h(x)h(y)h(z)=0$. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.

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.

Travelling wavefronts for a system of two reaction–diffusion equations are studied. The existence of a family of wavefronts (one for each wave speed) as well as the existence of the minimal speed (in the case that the Lewis number is greater than $1$) are proved. Asymptotic formulae for the wavefronts are established. New results are obtained when applying the main theorems to an isothermal autocatalytic chemical reaction system.

In certain rings containing non-central idempotents we characterize homomorphisms, derivations, and multipliers by their actions on elements satisfying some special conditions. For example, we consider the condition that an additive map $h$ between rings $\mathcal{A}$ and $\mathcal{B}$ satisfies $h(x)h(y)h(z)=0$ whenever $x,y,z\in\mathcal{A}$ are such that $xy=yz=0$. As an application, we obtain some new results on local derivations and local multipliers. In particular, we prove that if $\mathcal{A}$ is a prime ring containing a non-trivial idempotent, then every local derivation from $\mathcal{A}$ into itself is a derivation.

In this paper we study the quasi-static crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.

Riesz transforms $R_\mu$ related to the Bessel operators

$$ \Delta_\mu=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2} $$

are studied in this work. We develop for $R_\mu$ a theory that runs parallel to that for the Euclidean Hilbert transform. It is proved that $R_\mu$ is actually a Calderón–Zygmund singular integral operator. Also, $R_\mu$ is seen to be the boundary value of the appropriate harmonic extension for this context. Finally, we analyse weighted inequalities involving $R_\mu$.

It is shown that, if $f$ is a meromorphic function of order zero and $q\in\mathbb{C}$, then

\begin{equation} \label{abstid} m\bigg(r,\frac{f(qz)}{f(z)}\bigg)=o(T(r,f)) \tag{\ddag} \end{equation}

for all $r$ on a set of logarithmic density $1$. The remainder of the paper consists of applications of identity \eqref{abstid} to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of $q$-difference equations. The results obtained include $q$-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

We establish that, for $n\geq3$, the elliptic equation

$$ -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u $$

on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all $\lambda>0$ provided that

$$ \mu,\nu>-2,\quad\max\bigg\{2,\frac{n+2\mu+2}{n-2}\bigg\}<q<\frac{2(n+\mu)}{n-2}\quad\text{and}\quad p=\frac{2(n+\nu)}{n-2}. $$

When $q=2$ and $n>2\mu+6$, the same result holds for $\lambda>0$ small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation

$$ -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, $$

on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all $\tilde\lambda>0$ provided that $a,\nu>-2$ and

$$ \max\bigg\{2,\frac{n+a-\sqrt{\bar{\chi}-\chi}}{\sqrt{\bar{\chi}}}\bigg\} <q<\frac{n+a}{\sqrt{\bar{\chi}}}\quad\text{with~}\bar{\chi}=\bigg(\frac{n-2}{2}\bigg)^2. $$

When $q=2$, $n>2a+6$ and $0<\chi<\bar\chi-(a+2)^2$, the same result holds for $\tilde\lambda>0$ small.

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 study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits.

We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

We study the behaviour at infinity, with respect to the spatial variable, of solutions to the magnetohydrodynamics equations in $\mathbb{R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free non-stationary Navier–Stokes equations when $|x|\to\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are new boundedness criteria for convolution operators in weighted spaces.

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.

We find positive rapidly decaying solutions for the equation

$$ -\text{div}(K(x)\nabla u)=K(x)u^{2^*-1}+\lambda K(x)|x|^{\alpha-2}u $$

in $\mathbb{R}^N$, where $N\geq3$, the nonlinearity is given by the critical Sobolev exponent $2^*=2N/(N-2)$, the weight is $K(x)=\exp(\tfrac14|x|^\alpha)$, $\alpha\geq2$ and $\lambda$ is a parameter.

We are concerned with variational properties of a fold energy for a unit, dilation-invariant gradient field (called a cluster) in the unit disc. We show that boundedness of a fold energy implies $L^{1}$-compactness of clusters. We also show that a fold energy is $L^{1}$-lower semicontinuous. We characterize absolute minimizers. We also give a sequence of stationary states and discuss its stability. Surprisingly, the stability depends upon $q$, the power of modulus of the jump discontinuities, in the definition of the fold energy.

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 study the asymptotic behaviour of two nonlinear eigenvalue problems which involve $p$-Laplacian-type operators. In the first problem we consider the limit as $p\to\infty$ of the sequences of the $k$th eigenvalues of the $p$-Laplacian operators. The second problem we study is the homogenization of nonlinear eigenvalue problems for some $p$-Laplacian-type operators with $p$ fixed. Our asymptotic analysis relies on a convergence result for particular critical values of a class of Rayleigh quotients, stated in a unified framework, and on the notion of $\varGamma$-convergence.