Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

The question of parallelizability and stable parallelizability of a family of manifolds obtained as a quotient of circle action on the complex Stiefel manifolds is studied and settled.

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form

Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.

We study the Cauchy problem for the degenerate and uniformly parabolic equations with gradient term. The local existence, global existence and non-existence of solutions are obtained. In the case of global solvability, we get the exact estimates of a solution. In particular, we obtain the global existence of solutions in the limiting case.

This paper is concerned with the initial boundary-value problem for the p-system with nonlinear damping. We prove the existence of a global smooth solution under the assumption that only the C0-norm of the derivative of the initial data is sufficiently small, while the C0-norm of the initial data is not necessarily small. The proof is based on several key a priori estimates, the maximum principle and the characteristic method.

In this paper we investigate the differential geometry of 1-lightlike submanifolds in anti-de Sitter n-space as an application of the theory of Legendrian singularities. Based on some theory of lightlike submanifolds, we also introduce the notion of 1-lightlike horospherical Gauss curvature, which is important for us to study the singularities of 1-lightlike horospherical hypersurfaces. Moreover, we discuss the related geometric property of 1-lightlike horospherical hypersurfaces in anti-de Sitter n-space.

We consider the positive solutions to singular boundary-value problems of the form where λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

We analyse the multiplicity of non-negative solutions for the critical concave—convex-type equation

where Ω is a bounded domain of ℝN, 1 < r < p and λ > 0, and μ is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when λ is positive and sufficiently small and μ is in a right neighbourhood of μ1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2.

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limits

are positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

We obtain a solution, the L∞-norm of which is greater than ⅓√3, to the semilinear system arising in the mathematical theory of superconductivity when the domain is a disc or the exterior of a disc in ℝ2. We obtain an estimate of the critical field for the arbitrary penetration depth and determine the critical field as the penetration depth goes to zero when the domain is a disc. We prove that the L∞-norm of the symmetric solutions on the inner boundary is greater than that of the outer boundary when the domain Ω is an annulus. Stability of the solutions is also studied.

Let n ≥ 3 and let ψλ0 be the radially symmetric solution of Δ log ψ + 2βψ + βx · ∇ψ = 0 in ℝn, ψ(0) = λ(0), for some constants λ0 > 0, β > 0. Suppose u0 ≥ 0 satisfies u0 − ψλ0 ∈ L1 (ℝn) and u0 (x) ≈ (2(n − 2)/β)(log∣x∣/∣x∣2) as ∣x∣ → ∞. We prove that the rescaled solution ũ(x,t) = e2βtu(eβtx, t) of the maximal global solution u of the equation ut = Δ log u in ℝn × (0, ∞), u(x, 0) = u0 (x) in ℝn, converges uniformly on every compact subset of ℝn and in L1 (ℝn) to ψλ0 as t → ∞. Moreover, ∥ũ(·, t) − ψλ0∥L1(ℝn) ≤ e−(n−2)βt∥u0 − ψλ0∥L1(ℝn) for all t ≥ 0.

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.

This paper studies the asymptotic behaviour as α := u(0) ↑∞ of the first zero R(α) of the radially symmetric solution of the semilinear equation

in ℝn, n ≥ 1, where h > 0 and β > 1. We establish that R(α) = O(α−(β−1)/2) if n = 1, 2 or n ≥ 3 and β < (n + 2)/(n − 2), and conjecture that lim inf α→∞R(α) > 0 if n ≥ 3 and β > (n + 2)/(n − 2).

The period function of Liénard systems ẋ = −y + F(x), ẏ = g(x) associated with a centre is considered and sufficient conditions are presented for the period function to be strictly and globally monotone increasing and to have at least one critical point. Examples of applications and remarks on related work are also given.

We present new results in the theory of the classical theta functions of Jacobi: series expansions and defining ordinary differential equations (ODEs). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta functions; they also yield an exponential quadratic extension of the canonical θ-series. An integrability condition of these ODEs explains the appearance of the modular ϑ-constants and differential properties thereof. General solutions to all the ODEs are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences.

We derive a compatible condition between coefficients and negative exponents, which is optimal for H01-solutions of strongly singular problems. This allows us to establish the existence of H01-solutions for p ≥ 3. Our proof is based on a variational argument.