Browsing by Author "Chuaqui, M"

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    Characteristic properties of Nehari functions
    (PACIFIC JOURNAL MATHEMATICS, 1999) Chuaqui, M; Pommerenke, C
    Let N be the set of all meromorphic functions f defined in the unit disc D that satisfy Nehari's univalence criterion (1 - \z\(2))(2)\Sf(z)\ less than or equal to 2. In this paper we investigate certain properties of the class N. We obtain sharp estimates for the spherical distortion, and also a two-point distortion theorem that actually characterizes the set N. Finally, we study some aspects of the boundary behavior of Nehari functions, and obtain results that indicate how such maps can fail to map D onto a quasidisc.
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    Constant principal strain mappings on 2-manifolds
    (2000) Chuaqui, M; Gevirtz, J
    We study mappings between Riemannian 2-manifolds which have constant principal stretching factors (cps-mappings). Such mappings f can be described in terms of the relationship between the geodesic curvature of the curves of principal strain at p and that of their images at f (p). In the context of local coordinates this relationship takes the form of a nonlinear hyperbolic system, the blow-up properties of which depend on the Gaussian curvatures of the two manifolds. We use the theory of such systems to study global existence when both manifolds are the hyperbolic plane H-2 and obtain a simple description of all cps-mappings of H-2 onto itself. We also obtain a distortion result for disks in H-2 as well as some nonexistence results for cps-mappings of the Euclidean plane onto certain classes of manifolds. In addition, our treatment of cps-mappings in H-2 yields, virtually as a corollary, a generalization of a theorem of Epstein to the effect that a curve in hyperbolic n-space whose geodesic curvature is bounded by 1 must be simple.
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    Ellipses, near ellipses, and harmonic Mobius transformations
    (2005) Chuaqui, M; Duren, P; Osgood, B
    It is shown that an analytic function taking circles to ellipses must be a Mobius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Mobius transformation.
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    Finding complete conformal metrics to extend conformal mappings
    (INDIANA UNIV MATH JOURNAL, 1998) Chuaqui, M; Osgood, B
    This paper shows how new differential geometric approaches to univalence criteria involving the Schwarzian derivative can be applied to a classical, but very general, criterion of Nehari. We show how positive solutions to the second order ODE associated to the Schwarzian can be used to construct complete conformal metrics. These lead to explicit formulas for homeomorphic and quasiconformal extensions of conformal mappings as generalizations of the Ahlfors-Weill extension.
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    General univalence criteria in the disk: Extensions and extremal function
    (SUOMALAINEN TIEDEAKATEMIA, 1998) Chuaqui, M; Osgood, B
    Many classical univalence criteria depending on the Schwarzian derivative are special cases of a result, proved in [18], involving both conformal mappings and conformal metrics. The classical theorems for analytic functions on the disk emerge by choosing appropriate conformal metrics and computing a generalized Schwarzian. The results in this paper address questions of extending functions which satisfy the general univalence criterion; continuous extensions to the closure of the disk, and homeomorphic and quasiconformal extensions to the sphere. The main tool is the convexity of an associated function along geodesics of the metric. The other important aspect of this study is an extremal function associated with a given criterion, along with its associated extremal geodesics. An extremal function for a criterion is one whose image is not a Jordan domain. An extremal geodesic joins points on the boundary which map to the same point in the image. We show that, for the general criterion, the image of an extremal geodesic under an extremal function is a euclidean circle.
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    John domains, quasidisks, and the Nehari class
    (WALTER DE GRUYTER & CO, 1996) Chuaqui, M; Osgood, B; Pommerenke, C
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    On an elliptic problem with boundary blow-up and a singular weight
    (2003) Chuaqui, M; Cortázar, C; Elgueta, M; Flores, C; Letelier, R; García-Melián, J
    In this work we consider the non-autonomous problem Deltau = a(x)u(m) in the unit ball B subset of R-N, with the boundary condition u\(partial derivativeB) = +infinity, and m > 0. Assuming that a is a continuous radial function with a(x) similar to C-0 dist(x, partial derivativeB)(-gamma) as dist(x, partial derivativeB) --> 0, for some C-0 > 0, gamma > 0, we completely determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of m and gamma. The case 0 < m less than or equal to 1, as well as estimates for solutions to the linear problem m = 1, are a significant part of our results.
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    Simple curves in Rn and Ahlfors' Schwarzian derivative
    (2004) Chuaqui, M; Gevirtz, J
    We derive sharp injectivity criteria for mappings f : ( 1; 1) --> R-n in terms of Ahlfors' definition of the Schwarzian derivative for such mappings.
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    The Schwarzian derivative for harmonic mappings
    (2003) Chuaqui, M; Duren, P; Osgood, B
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    Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights
    (2004) Chuaqui, M; Cortazar, C; Elgueta, M; Garcia-Melian, J
    We consider the elliptic problems Deltau = a(x)u(m), m > 1, and Deltau = a(x)e(u) in a smooth bounded domain Omega, with the boundary condition u = +infinity on partial derivativeOmega. The weight function a(x) is assumed to be Holder continuous, growing like a negative power of d(x) = dist(x, partial derivativeOmega) near partial derivativeOmega. We show existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.