Mostrando recursos 1 - 13 de 13

  1. Spectral analysis of the subelliptic oblique derivative problem

    Taira, Kazuaki
    This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter $\lambda$. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when $\lvert \lambda \rvert$ tends to $\infty$. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of $L^2$ Sobolev spaces.

  2. A note on approximation of plurisubharmonic functions

    Persson, Håkan; Wiegerinck, Jan
    We extend a recent result of Avelin, Hed, and Persson about approximation of functions $f$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\overline{\Omega}$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\overline{\Omega}$. We show that such approximation is possible if the boundary of $\Omega$ is $C^0$ outside a countable exceptional set $E \subset \partial \Omega$. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the...

  3. Invertibility of nonsmooth mappings

    Montenegro, Marcelo; Presoto, Adilson E.
    Let $F : \mathbb{R}^N \to \mathbb{R}^N$ be a locally Lipschitz continuous function. We prove that $F$ is a global homeomorphism or only injective, under suitable assumptions on the subdifferential $\partial F(x)$. We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard–Levy Theorem. We also address questions on the Markus–Yamabe conjecture.

  4. A characterization of Herglotz–Nevanlinna functions in two variables via integral representations

    Luger, Annemarie; Nedic, Mitja
    We derive an integral representation for Herglotz–Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.

  5. A four-dimensional Neumann ovaloid

    Karp, Lavi; Lundberg, Erik
    What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as...

  6. Measures with predetermined regularity and inhomogeneous self-similar sets

    Käenmäki, Antti; Lehrbäck, Juha
    We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set $C$. If the Assouad dimension of $C$ is strictly...

  7. Equivariant $L^2$-Euler characteristics of $G\textrm{-}CW$-complexes

    Jo, Jang Hyun
    We show that if $X$ is a cocompact $G\textrm{-}CW$-complex such that each isotropy subgroup $G_\sigma$ is $L^{(2)}$-good over an arbitrary commutative ring $k$, then $X$ satisfies some fixed-point formula which is an $L^{(2)}$-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper $G\textrm{-}CW$-complex which relates the equivariant $L^{(2)}$-Euler characteristic of a fixed point $CW$-complex $X^s$ and the Euler characteristic of $X/G$. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji–Mislin in 2009. We also show that if X is a free $G\textrm{-}CW$-complex such...

  8. Approximations and examples of singular Hermitian metrics on vector bundles

    Hosono, Genki
    We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank $2$ bundles on elliptic curves.

  9. Modulus in Banach function spaces

    Honzlová Exnerová, Vendula; Malý, Jan; Martio, Olli
    Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz $L^{p,1}$-space for an appropriate $p$, in particular, to the ordinary $L^1$-space if $p=1$, but not by the ordinary modulus with respect to the same space.

  10. The behavior of depth functions of cover ideals of unimodular hypergraphs

    Hang, Nguyen Thu; Trung, Tran Nam
    We prove that the cover ideals of all unimodular hypergraphs have the non-increasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.

  11. Algebraic independence of the values of power series with unbounded coefficients

    Hajime, Kaneko
    Many mathematicians have studied the algebraic independence over $\mathbb{Q}$ of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over $\mathbb{Q}$ of the values $\sum^{\infty}_{n=0} t(n) \beta^{-n}$ for distinct sequences $(t(n))^{\infty}_{n=0}$ of nonnegative integers, where $\beta$ is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values $\sum^{\infty}_{n=0} t_1 (n) \beta^{-n} , \dotsc , \sum^{\infty}_{n=0}...

  12. The Auslander bijections and universal extensions

    Chen, Xiao-Wu
    Universal extensions arise naturally in the Auslander bijections. For an abelian category having Auslander–Reiten duality, we exploit a bijection triangle, which involves the Auslander bijections, universal extensions and the Auslander–Reiten duality. Some consequences are given, in particular, a conjecture by Ringel is verified.

  13. Hölder regularity for degenerate parabolic obstacle problems

    Bögelein, Verena; Lukkari, Teemu; Scheven, Christoph
    We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous.

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