• Title/Summary/Keyword: Liouville theorem

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TAMED EXHAUSTION FUNCTIONS AND SCHWARZ TYPE LEMMAS FOR ALMOST HERMITIAN MANIFOLDS

  • Weike, Yu
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.6
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    • pp.1423-1438
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    • 2022
  • In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish a related Schwarz type lemma for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce several versions of Schwarz and Liouville type theorems for almost holomorphic maps.

AN INVESTIGATION ON THE EXISTENCE AND UNIQUENESS ANALYSIS OF THE FRACTIONAL NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

  • Fawzi Muttar Ismaael
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.1
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    • pp.237-249
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    • 2023
  • In this paper, by means of the Schauder fixed point theorem and Arzela-Ascoli theorem, the existence and uniqueness of solutions for a class of not instantaneous impulsive problems of nonlinear fractional functional Volterra-Fredholm integro-differential equations are investigated. An example is given to illustrate the main results.

ROUGH ISOMETRY AND HARNACK INEQUALITY

  • Park, Hyeong-In;Lee, Yong-Hah
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.455-468
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    • 1996
  • Certain analytic behavior of geometric objects defined on a Riemannian manifold depends on some very crude properties of the manifold. Some of those crude invariants are the volume growth rate, isoperimetric constants, and the likes. However, these crude invariants sometimes exercise surprising control over the analytic behavior.

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SINGULARITY ESTIMATES FOR ELLIPTIC SYSTEMS OF m-LAPLACIANS

  • Li, Yayun;Liu, Bei
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1423-1433
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    • 2018
  • This paper is concerned about several quasilinear elliptic systems with m-Laplacians. According to the Liouville theorems of those systems on ${\mathbb{R}}^n$, we obtain the singularity estimates of the positive $C^1$-weak solutions on bounded or unbounded domain (but it is not ${\mathbb{R}}^n$ and their decay rates on the exterior domain when ${\mid}x{\mid}{\rightarrow}{\infty}$. The doubling lemma which is developed by Polacik-Quittner-Souplet plays a key role in this paper. In addition, the corresponding results of several special examples are presented.

ANALYTIC SOLUTION OF HIGH ORDER FRACTIONAL BOUNDARY VALUE PROBLEMS

  • Muner M. Abou Hasan;Soliman A. Alkhatib
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.3
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    • pp.601-612
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    • 2023
  • The existence of solution of the fractional order differential equations is very important mathematical field. Thus, in this work, we discuss, under some hypothesis, the existence of a positive solution for the nonlinear fourth order fractional boundary value problem which includes the p-Laplacian transform. The proposed method in the article is based on the fixed point theorem. More precisely, Krasnosilsky's theorem on a fixed point and some properties of the Green's function were used to study the existence of a solution for fourth order fractional boundary value problem. The main theoretical result of the paper is explained by example.

ROUGH ISOMETRY, HARMONIC FUNCTIONS AND HARMONIC MAPS ON A COMPLETE RIEMANNIAN MANIFOLD

  • Kim, Seok-Woo;Lee, Yong-Han
    • Journal of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.73-95
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    • 1999
  • We prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincar inequality and the finite covering condition at infinity on each end, then every positive harmonic function on the manifold is asymptotically constant at infinity on each end. This result is a direct generalization of those of Yau and of Li and Tam.

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AN ENERGY DENSITY ESTIMATE OF HEAT EQUATION FOR HARMONIC MAP

  • Kim, Hyun-Jung
    • The Pure and Applied Mathematics
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    • v.18 no.1
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    • pp.79-86
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    • 2011
  • Suppose that (M,g) is a complete and noncompact Riemannian mani-fold with Ricci curvature bounded below by $-K{\leq}0$ and (N, $\bar{g}$) is a complete Riemannian manifold with nonpositive sectional curvature. Let u : $M{\times}[0,{\infty}){\rightarrow}N$ be the solution of a heat equation for harmonic map with a bounded image. We estimate the energy density of u.

ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS AND EXPONENTIALLY HARMONIC FUNCTIONS

  • Chi, Dong-Pyo;Choi, Gun-Don;Chang, Jeong-Wook
    • Journal of the Korean Mathematical Society
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    • v.39 no.5
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    • pp.731-743
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    • 2002
  • Let M be a Riemannian manifold with asymptotically non-negative curvature. We study the asymptotic behavior of the energy densities of a harmonic map and an exponentially harmonic function on M. We prove that the energy density of a bounded harmonic map vanishes at infinity when the target is a Cartan-Hadamard manifold. Also we prove that the energy density of a bounded exponentially harmonic function vanishes at infinity.

FOURIER'S TRANSFORM OF FRACTIONAL ORDER VIA MITTAG-LEFFLER FUNCTION AND MODIFIED RIEMANN-LIOUVILLE DERIVATIVE

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1101-1121
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    • 2008
  • One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

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A LIOUVILLE TYPE THEOREM FOR HARMONIC MORPHISMS

  • Jung, Seoung-Dal;Liu, Huili;Moon, Dong-Joo
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.941-947
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    • 2007
  • Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let ${\mu}0$ be the least eigenvalue of the Laplacian acting on $L^2-functions$ on M. We show that if $Ric^M{\ge}-{\mu}0$ at all $x{\in}M$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism ${\phi}:M{\to}N$ of finite energy is constant.