Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems

A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.     

Uniqueness of positive solutions of Δu−u+up=0 in Rn

We establish the uniqueness of the positive, radially symmetric solution to the differential equation u–u+u^p=0 (with p>1) in a bounded or unbounded annular region in R ⁿ for all n1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.     

A group theoretical approach to a certain class of perturbations of the Kepler problem

We study a particular class of perturbations of the classical Kepler Hamiltonian, first in two, then in three and finally in n dimensions. At every stage of our investigation the group theoretical nature of our constructions is fully exposed.In particular we present a new regularization of the n-dimensional Kepler problem which is based on previous constructions of Guillemin & Sternberg (see [8]). This regularization is similarily related to Moser's (see [9]) as is Kustaanheimo-Stiefel's (see [4]) in three dimensions.     

Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

A principle of objectivity for relativistic continuum mechanics

A principle of objectivity is formulated for Galilei and Minkowski space-time structure. In the Galilei case it is the principle of Noll. A reduced form for arbitrary local and causal functionals is obtained. For materials of grade n it coincides with the result of Bressan, and for simple materials with that of Bragg.     

Best constants in Sobolev inequalities for ultraspherical polynomials

A family of sharp Sobolev-type inequalities for functions on the classical measure spaces associated with the ultraspherical or Gegenbauer polynomials is obtained. These estimates generalize the Sobolev inequalities for the n-sphere S ⁿ given by Beckner, and are derived from a sharp Sobolev inequality for functions on the real line. Spectral considerations allow these estimates to be expressed as multiplier inequalities for functions which have expansions in terms of Gegenbauer polynomials.     

Solitons in Several Space Dimensions:¶Derrick's Problem and¶Infinitely Many Solutions

In this paper we study a class of Lorentz invariant nonlinear field equations in several space dimensions. The main purpose is to obtain soliton-like solutions. These equations were essentially proposed by C. H. Derrick in a celebrated paper in 1964 as a model for elementary particles. However, an existence theory was not developed. The fields are characterized by a topological invariant, the charge. We prove the existence of a static solution which minimizes the energy among the configurations with nontrivial charge. Moreover, under some symmetry assumptions, we prove the existence of infinitely many solutions, which are constrained minima of the energy. More precisely, for every n∈:N there exists a solution of charge n. Accepted March 13, 2000?Published online September 12, 2000     

Some theorems on holomorphic matrices

We consider holomorphic m x m-matrices, i.e. m x m-matrices whose matrix elements are holomorphic functions on a connected open set G in the space C_n of n complex variables.After giving some physical motivation for this research we discuss the analytic behavior of the eigenvalues, the construction of a G-adherent transformation matrix and a (physically) useful commutator theory. Vorgelegt von H. Görtler     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

Existenzbeweis für cnoidal waves unter berücksichtigung der oberflächenspannung

Ohne ZusammenfassungHerrn Prof. Herbert Beckert zum 50. Geburtstag Vorgelegt von J. Serrin     

Fit regions and functions of bounded variation

Dedicated to James B. Serrin on the occasion of his sixtieth birthday     

Generalized characteristics in hyperbolic systems of conservation laws

In Memoriam Ronald J. DiPerna     

The Bianchi identities in an explicit form

Dedicated to Professor Bernard D. Coleman     

Avere una memoria tenace crea gravi problemi

Dedicata a Clifford A. Truesdell nel Suo 60 compleanno     

Die polynären Thermodiffusionskoeffizienten von Gasgemischen

Zusammenfassung Für die polynären ThermodiffusionskoeffizientenD _i ^T von Mehrkomponentengemischen aus einatomigen Gasen geringer Dichte wird eine im Sinne der ersten Chapman-Enskog-Approximation exakte Gleichung abgeleitet. Im Gegensatz zu dem vonHirschfelder et al. angegebenen Ausdruck besitzt die neue Formel eine einfache mathematische Form, die die Thermodiffusion in einem binären Gemisch als Grenzfall beschreibt. DieD _i ^T werden mit den polynären Thermodiffusionsfaktoren verknüpft. Weiterhin wird angegeben, wie die neue Formel auch für Gemische aus mehratomigen Gasen verwendet werden kann. Als Beispiel werden dieD _i ^T von dissoziierter Luft, die als Fünf-Komponenten-Gemisch behandelt wird, ermittelt. Abschließend wird gezeigt, daß die Berechnung derD _i ^T durch die Verwendung genauer Näherungsbeziehungen für die Wärmeleitfähigkeit von Mehrkomponentengemischen erheblich vereinfacht werden kann.

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The multicomponent thermal diffusion coefficients of gas mixtures

For low density mixtures of monatomic gases, an equation is derived for the multicomponent thermal diffusion coefficientsD _i ^T . The equation is exact in the meaning of the first Chapman-Enskog-approximation. In contrast to the well-known Hirschfelder-expression for theD _i ^T , the new formula has a simple mathematical structure which does not fail to be reduced to the proper form for a binary mixture. A relation between theD _i ^T and the multicomponent thermal diffusion factors is presented. Moreover the equation is justified for the evaluation of theD _i ^T of polyatomic, multicomponent mixtures. As an example, theD _i ^T are calculated for dissociated equilibrium air as a five-component system. Finally, it is demonstrated that by use of accurate approximations known for multicomponent heat conductivity coefficients, a more readily tractable form of the new equation can be obtained for computational purposes.

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Bezeichnungen D binärer Diffusionskoeffizient - D ^T polynärer Thermodiffusionskoeffizient - j Diffusionsstromvektor - m Molekülmasse - M Molmasse - n Teilchendichte - p statischer Druck - P Drucktensor - q Energiestromvektor - T absolute Temperatur - V Diffusionsgeschwindigkeit - x Molenbruch - polynärer Thermodiffusionsfaktor - Zähigkeit - Wärmeleitfähigkeit - Massendichte Der Verfasser dankt den Herren Dr. E.Adams, Dr. A.Schabee, Dr. W.Schönauer und Dipl.-Math. L.Wassner für viele wertvolle Anregungen. Die Arbeit wurde von der Deutschen Forschungsgemeinschaft dankenswerterweise finanziell unterstützt.     

On the equilibrium of a cylindrical elastic solid: Comparisons with St. Venant's theory

Communicated by M. E. Gurtin &; C. Truesdell     

Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit

 We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195–211]. (Accepted June 6, 2002) Published online December 3, 2002 Communicated by Y. BRENIER     

Zur mathematischen theorie akustischer Wellenfelder

Ohne Zusammenfassung Vorgelegt von C. Müller Zur praktischen Berechnung von U können nach Satz 2 die numerischen Methoden zur Auflösung Fredholmscher Integralgleichungen, etwa das Verfahren von Cl. Müller [10], verwendet werden.     

Über den Resonanz fall bei Systemen von n nichtlinearen gewöhnlichen Differentialgleichungen erster Ordnung

Ohne Zusammenfassung Vorgelegt von L. Collatz II. Teil einer bei der Naturwissenschaftlich-Philosophischen Fakultät der Technischen Hochschule zu Braunschweig eingereichten Dissertation. Referenten: Prof. Dr. R. Iglisch und Prof. Dr. H.-J. Kanold. Auch Herrn Dr. H. Eltermann bin ich für wertvolle Ratschläge zu großem Dank verpflichtet.     

On patches of uniform vorticity in a plane of irrotational flow

We derive an evolution equation for the motions of patches of vorticity (vortex). Steady state solutions of this equation that include those of Kirchhoff and Moore & Saffman are established. The m-fold symmetric, m3, hypotrochoid is an exact steady solution of this equation when rotation and strain are present. When strain is absent but rotation is present, the m-fold symmetric, m2, hypotrochoid is a perturbation solution with a dispersion relation extending that of Lamb. The case of m=2 is exact and is the Kirchhoff elliptical vortex.