Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

On applications of the Taylor formula in some quasispaces

We consider some metric spaces with quasimetric (quasispaces) comprising uniformly regular (equiregular) Carnot — Carathéodory quasispaces whose quasimetric is induced by C ^ϒ−1-smooth vector fields of formal degree not higher than ϒ. For these spaces, some analogues of the Campbell — Hausdorff formula are derived, which allows us to prove a theorem on a nilpotent tangent cone, a theorem on isomorphism of various nilpotent tangent cones defined at a common point, and a local approximation theorem.     

Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary

Abstract

In this work, we study the Carathéodory approximate solution for a class of one-dimensional perturbed stochastic differential equations with reflecting boundary (PSDERB). Based on the Carathéodory approximation procedure, we prove that PSDERB have a unique solution and show that the Carathéodory approximate solution converges to the solution of PSDERB whose both drift and diffusion coefficients are non-Lipschitz. After that, we establish an explicit rate of convergence in the case of PSDERB with Lipschitz coefficients.     

The Carathéodory topology for multiply connected domains I

We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.     

Analytic Description of the Spaces Dual to Spaces of Holomorphic Functions of Given Growth on Carathéodory Domains

The spaces dual to spaces of holomorphic functions of given growth on Carathéodory domains are described by using the Cauchy transform of functionals. A pseudoanalytic extension of such transforms to the whole plane is constructed, which makes it possible to remove convexity constrains and consider spaces determined by weights of general form, rather than only by those whose dependence on the distance from a point of the domain to its boundary is one-dimensional.     

Banach空间二阶非线性微分方程的弱Carathéodory解

本文定义了二阶微分方程的弱 Carathéodory解 ,在不涉及紧型条件的情形下 ,直接用迭代法证明了 Banach空间二阶非线性常微分方程两点边值问题存在唯一解 ,并给出逼近解迭代序列的误差估计 ,对周期边值问题得到类似的结果     

On Ranges and Moore�CPenrose Inverses Related to Matrix Carath��odory and Schur Functions

By Cayley transformation, there is an interplay between matrix-valued Carathéodory and Schur functions in the unit disk. One of the main aims of the paper is to study this interplay with a view to the fact that the matrix function given by the values of a Carathéodory function via Moore–Penrose inverses is also a Carathéodory function. In doing so, we also analyze ranges and null spaces of the values of the matrix function in question and get some results of independence concerning the concretely chosen point of the domain.     

On the Schröder-Bernstein problem for carathéodory vector lattices

In this note we prove that there exists a Carathéodory vector lattice V such that V ≌ V ³ and V ?V ². This yields that V is a solution of the Schröder-Bernstein problem for Carathéodory vector lattices. We also show that no Carathéodory Banach lattice is a solution of the Schröder-Bernstein problem.     

The Hadamard Theorem and Borel–Carathéodory Theorem on Riemann Surfaces

In this paper, we first define a kind of pseudo–distance function and annulus domain on Riemann surfaces, then prove the Hadamard Theorem and the Borel–Carathéodory Theorem on any Riemann surfaces. Supported by NSFC 10501052     

Notes About the Carathéodory Number

In this paper we give sufficient conditions for a compactum in ? ⁿ to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.     

Upper bound for the Lempert function of smooth domains

An upper estimate for the Lempert function of any C ^1+ε -smooth bounded domain in \mathbbCⁿ{\mathbb{C}^n} is found in terms of the boundary distance.     

The Lempert Theorem and the Tetrablock

In this paper we show that the Lempert property (i.e., the equality between the Lempert function and the Carathéodory distance) holds in the tetrablock, a bounded hyperconvex domain which is not biholomorphic to a convex domain. The question whether such an equality holds was posed by Abouhajar et al. in J. Geom. Anal. 17(4), 717–750 (2007).     

Nonlinear Elliptic Equations with Singular Terms and Combined Nonlinearities

We consider nonlinear elliptic Dirichlet problems with a singular term, a concave (i.e., (p − 1)-sublinear) term and a Carathéodory perturbation. We study the cases where the Carathéodory perturbation is (p − 1)-linear and (p − 1)-superlinear near +∞. Using variational techniques based on the critical point theory together with truncation arguments and the method of upper and lower solutions, we show that if the L ^∞-coefficient of the concave term is small enough, the problem has at least two nontrivial smooth solutions.     

Ereditarietà delle misure di Caratheodory

Let μ be a diffuse Carathéodory measure. The purpose of this paper is to prove that, for every open setU inR, μ(·∩U) is a Carathéodory measure too.     

Boundary behaviour of invariant functions on planar domains

Precise behaviour of the Carathéodory, Kobayashi and Bergman metrics and distances near smooth boundary points of planar domains is found under various assumptions of regularity.     

A topological principle for retarded functional differential equations of Carathéodory type

The paper presents a Razumikhin-type version of Wa?ewski's principle for RFDEs of Carathéodory type, thus generalizing the author's previous results (J. Differential Equations36, 117–138 (1980)). A few facts from the theory of RFDEs are reviewed, and the concept of a regular polyfacial set with respect to an RFDE of Carathéodory type is introduced and the results are state and proved. Most of the examples given in the above work can easily be adapted to the more general setting of Carathéodory systems. An indication how to do it is given in only one case. Throughout the paper, standard notation is used.     

Bad boundary behavior in star-invariant subspaces I

We discuss the boundary behavior of functions in star-invariant subspaces (BH ²)^⊥, where B is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of B where B does not admit a derivative in the sense of Carathéodory.     

Carathéodory balls and norm balls of the domainH={(z1,z2)∈C2∶|z1|+|z2|<1}

LetH be the domain inC ² defined byH={Z=(z ₁,z ₂):║Z║₁=│z║₁│+│z║₂│<1}. LetC _H(z,w) be the Carathéodory distance ofH,z,w∈H. The Carathéodory ballB _C(z_C,α;H) with centerz _C,z_C∈H, and radius α, 0<α<1, is defined byB _c(z_C,α;H)={z∶C_H(z,z_C)<arc tanh α}. The norm ballB _N(z_N,r) with centerz _N,z_N∈H, and radiusr, 0<r<1-‖z _N‖₁, is defined byB _N(z_N,r)={z∶ ‖z−z_N‖₁<r}. Theorem:The only Carathéodory balls of H which are also norm balls are those with their center at the origin.     

Strong Solvability of a Unilateral Boundary Value Problem for Nonlinear Parabolic Operators

Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators. The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with respect to the independent variable X is required. The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary conditions and the monotonicity of the operator − u _t with respect to Δu for the same functions.