On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

Boundedness of Stein’s spherical maximal function in variable Lebesgue spaces and application to the wave equation

If ${p(\cdot): \mathbb{R}^{n} {\rightarrow} (0,\infty), n\geq 3}$ , is globally log-Hölder continuous and its infimum p ^? and its supremum p ⁺ are such that ${\frac{n}{n-1} < p^{-} \leq p^{+} < p^{-} (n-1)}$ , then the spherical maximal operator (integral averages taken with respect to the (n ? 1)-dimensional surface measure) is bounded. When n = 3, the result is then interpreted as the preservation of the integrability properties of the initial velocity of propagation to the solution of the initial-value problem for the wave equation.     

New amino acid based biodegradable poly(ester amide)s via bis-azlactone chemistry

Abstract

Three new classes of the amino acid based biodegradable (AABB) polymers were synthesized via step growth polymerization of bis-azlactones and amino acid based diamine-diesters with activated fatty diester and alkylenediamine: a) poly(ester amide)s (PEAs) were obtained by polymerization of bis-azlactones with diamine-diesters, b) hydrophobically modified co-poly(ester amide)s (co-PEAs) were synthesized by copolymerization of activated fatty diacid diester and bis-azlactones with diamine-diesters, and c) poly(ester amide-co-amide)s (PEA-co-PAs) were obtained by copolymerization of alkylene diamine and diamine-diesters with bis-azlactones. The new poly(ester amide)s showed relatively low-molecular-weights (M_w within 2,800–19,600?Da, GPC in DMF), whereas the new co-poly(ester amide)s and poly(ester amide-co-amide)s exhibited high-molecular-weights (M_w within 40–100?kDa) leading to good mechanical properties. Incorporation of the bis-azlactone fragments into the poly(ester amide)s backbone increased hydrophobicity and thermal stability, whereas incorporation of diamine-diester units into the backbone of the bis-azlactone based polyamides rendered them biodegradable. Synthesized AABB polymers are potential candidates for constructing resorbable surgical and pharmaceutical devices.     

Computation of poisson type integrals

We consider problems of computing the Poisson integral when the point at which the integral is evaluated approaches the ball surface. Techniques are proposed enabling one to improve the computation efficiency.     

Mixed Boundary Value Problems of Thermopiezoelectricity for Solids with Interior Cracks

We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. This research was supported by the Georgian National Science Foundation grant GNSF/ST07/3-170 and by the German Research Foundation grant DFG 436 GEO113/8/0-1.     

Solvability and regularity results to boundary‐transmission problems for metallic and piezoelectric elastic materials

We investigate three‐dimensional transmission problems related to the interaction of metallic and piezoelectric ceramic bodies. We give a mathematical formulation of the physical problem when the metallic and ceramic sub‐domains are bonded along some proper parts of their boundaries. The corresponding nonclassical mixed boundary‐transmission problem is reduced by the potential method to an equivalent nonselfadjoint strongly elliptic system of pseudo‐differential equations on manifolds with boundary. We investigate the solvability of this system in different function spaces. On the basis of these results we prove uniqueness and existence theorems for the original boundary‐transmission problem. We study also the regularity of the electrical and mechanical fields near the curves where the boundary conditions change and where the interfaces intersect the exterior boundary. The electrical and mechanical fields can be decomposed into singular and more regular terms near these curves. A power of the distance from a reference point to the corresponding edge‐curves occurs in the singular terms and describes the regularity explicitly. We compute these complex‐valued exponents and demonstrate their dependence on the material parameters (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation theorems for variable exponent Lebesgue spaces

When Hardy-Littlewood maximal operator is bounded on Lp(⋅)(Rn) space we prove θ[Lp(⋅)(Rn),BMO(Rn)]=Lq(⋅)(Rn) where q(⋅)=p(⋅)/(1−θ) and θ[Lp(⋅)(Rn),H¹(Rn)]=Lq(⋅)(Rn) where 1/q(⋅)=θ+(1−θ)/p(⋅).     

Laplace–Beltrami equation on hypersurfaces and Γ‐convergence

A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface with the boundary is investigated. The main objective is to trace what happens in Γ‐limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ‐limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space . Copyright © 2017 John Wiley & Sons, Ltd.     

Boundary value problems of electroelasticity with concentrated singularities

We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighborhood of infinity or in the neighborhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.     

Gagliardo-Nirenberg type inequality for variable exponent Lebesgue spaces

We prove analogies of the classical Gagliardo-Nirenberg inequalities