We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

We study ribbons of vanishing Gaussian curvature, i.e. flat ribbons, constructed along a curve in $\mathbb {R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the difference between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every f ∈ LΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.

Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

We determine the critical coupling of the two-dimensional Brown–Ravenhall operator with Coulomb potential. Boundedness from below has essentially been proven by Bouzouina. However, that work contains a trivial error leading to an incorrect constant that is exactly half of the actual critical constant. Furthermore, we show that the operator is in fact positive. Our proof of that is, for the most part, analogous to Tix's proof of the corresponding result for the three-dimensional operator.

In this work we study the solvability of the initial boundary-value problems that model the quasi-static nonlinear behaviour of ferroelectric materials. Similar to the metal plasticity, the energy functional of a ferroelectric material can be additively decomposed into reversible and remanent parts. The remanent part associated with the remanent state of the material is assumed to be a convex non-quadratic function f of internal variables. In this work we introduce the notion of the measure-valued solutions for the ferroelectric models, and show their existence in the rate-dependent case, assuming the coercivity of the function f. Regularizing the energy functional by a quadratic positive-definite term, which can be viewed as hardening, we show the existence of measure-valued solutions for the rate-independent and rate-dependent problems, avoiding the coercivity assumption on f.

We consider a class of Schrödinger operators on ${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in $L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential:

1. (P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;

2. (P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set.

Using a standard theory of differential operators in Lebesgue spaces, we re-prove and generalize some results of Chernyavskaya and Shuster, giving (mostly sufficient) conditions that minimal operators determined by expressions of the form −(ry′)′ + qy with domain and range in possibly different Lp spaces on intervals with at least one singular endpoint have bounded inverses.

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

Let m(λ) denote the Titchmarsh–Weyl m-function associated with the differential equation

with Neumann boundary condition at 0. In the case q ∊ CM[0, ε) for some integer M and ε > 0 we prove the following result (Theorem 2.1): If for some integer ŋ ≧ 3 we know that Im {λ(ŋ + l)/2m0(λ)} = 0 on the ray λ = |λ| eiπ/ŋ, then q(v) (0) = 0 for all v in the set {0, 1, … M} for Which ≢ ŋ − 2 mod ŋ.

A description is given of the rational functions A(X), B(X) over a field Ω for which A(B(X)) is an nth power in Ω(X).