We present a new necessary condition for similarity of indefinite Sturm–Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl–Titchmarsh $m$-functions. We also obtain necessary conditions for regularity of the critical points $0$ and $\infty$ of $J$-non-negative Sturm–Liouville operators. Using this result, we construct several examples of operators with the singular critical point $0$. In particular, it is shown that $0$ is a singular critical point of the operator

$$-\frac{(\mathrm{sgn} x)}{(3|x|+1)^{-4/3}}\frac{\mathrm{d}^2}{\mathrm{d} x^2}$$

acting in the Hilbert space $L^2(\mathbb{R},(3|x|+1)^{-4/3}\,\mathrm{d} x)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a $J$-non-negative Sturm–Liouville operator of type $(\mathrm{sgn} x)(-\mathrm{d}^2/\mathrm{d} x^2+q(x))$ with the same properties.

Using Leray–Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation we prove an Ambrosetti–Prodi-type result for periodic solutions of the telegraph equation.

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions $(t,1/t,s)$ within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps $L\log^2L$ into $L^{1,\infty}$. We give a simple geometric condition that guarantees that the $L\log^2L$ class cannot be enlarged. The proof develops the author's methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fefferman and Pipher.

We prove weighted norm inequalities for vector-valued commutators of multilinear operators. We first consider the strong $(p,p)$ case with $p>0$ and then the case weak-type estimate.

Let $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

We introduce the concept of causality into the framework of generalized pseudo-Riemannian geometry in the sense of Colombeau and establish the inverse Cauchy–Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward by Hawking and Ellis. Our work is based on a new characterization of free elements in finite-dimensional modules over the ring of generalized numbers.

This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

We study biminimal immersions: that is, immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler–Lagrange equation associated with biminimal immersions for both biminimal curves in a Riemannian manifold, with particular attention given to the case of curves in a space form, and isometric immersions of codimension 1 in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods of constructing families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.

We compare solutions of a class of degenerate parabolic equations on a Riemannian manifold $M$ with solutions of the equation on a model manifold. The class of equations under consideration contains both the parabolic $p$-Laplace equation and the porous medium equation. We prove that, under curvature conditions, solutions on model manifolds induce sub- or supersolutions on $M$. Using this result, we obtain curvature-dependent estimates for the speed of propagation of solutions.

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore, we discuss the relations of such a Bloch-type condition with the analogous hyperbolic Lipschitz condition.

In this paper we construct solutions which develop two negative spikes as $\varepsilon\to0^+$ for the problem $-\Delta u=|u|^{4/(N-2)}u+\varepsilon f(x)$ in $\varOmega$, $u=0$ on $\partial\varOmega$, where $\varOmega\subset\mathbb{R}^N$ is a bounded smooth domain exhibiting a small hole, with $f\geq0$, $f\not\equiv0$. This result extends a recent work of Clapp et al. in the sense that no symmetry assumptions on the domain are required.

For $0\ltp\le1$, let $h^p(\mathbb{R}^n)$ denote the local Hardy space. Let $\mathcal{F}$ be a Fourier integral operator defined by the oscillatory integral

$$ \mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi, $$

where $\phi$ is a $\mathcal{C}^\infty$ non-degenerate real phase function, and $b$ is a symbol of order $\mu$ and type $(\rho,1-\rho)$, $\sfrac12\lt\rho\le1$, vanishing for $x$ outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$ and $\mu\le-(n-1)(1/p-1/2)$ then $\mathcal{F}$ initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$ extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$ and $\mu$ is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$ spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces.

Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry.

Asymptotic approximations to the Green's functions of Sturm–Liouville boundary-value problems on graphs are obtained. These approximations are used to study the regularized traces of the differential operators associated with these boundary-value problems. Various inverse spectral problems for Sturm–Liouville boundary-value problems on graphs resembling those considered in Halberg and Kramer's ‘A generalization of the trace concept' (Duke Mathematics Journal27 (1960), 607–617), for Sturm–Liouville problems, and Pielichowski's ‘An inverse spectral problem for linear elliptic differential operators' (Universitatis Iagellonicae Acta Mathematica27 (1988), 239–246), for elliptic boundary-value problems, are solved.

We study an implicit second-order ordinary differential equation with complete integral. The class of second-order Clairaut-type equations is an important class of completely integrable equations. Indeed, the notion of second-order Clairaut-type equations is the generalization of second-order classical Clairaut equations. We give generic classifications of second-order Clairaut-type equations under the weak equivalence and the one-parameter strict equivalence relations.

We establish the persistence of the asymptotic stability of a linear equation $v'=A(t)v$ in a Banach space under sufficiently small perturbations, when the linear equation admits a non-uniform exponential contraction or a non-uniform exponential dichotomy. Moreover, we obtain optimal estimates for the decay of solutions of the perturbed equation, that in general may depend on the initial time.

Corresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality

$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$

is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

In this paper, we investigate the existence of positive solutions to a second-order Sturm–Liouville boundary-value problem with impulsive effects. The ideas involve differential inequalities and variational methods.