• Title, Summary, Keyword: Liouville theorem

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RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Anastassiou, George A.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1423-1433
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    • 2019
  • Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

A LIOUVILLE-TYPE THEOREM FOR COMPLETE RIEMANNIAN MANIFOLDS

  • Choi, Soon-Meen;Kwon, Jung-Hwan;Suh, Young-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.301-309
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    • 1998
  • The purpose of this paper is to give a theorem of Liouvilletype for complete Riemannian manifolds as an extension of the Theorem of Nishikawa [6].

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LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

  • Jung, Seoung Dal
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.155-161
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    • 2014
  • Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that $Ric^M{\geq}-\frac{4(p-1)}{p^2}{\mu}_0$ at all $x{\in}M$ and Vol(M) is infinite, where ${\mu}_0$ > 0 is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M. Then any p-harmonic map ${\phi}:M{\rightarrow}N$ of finite p-energy is constant Also, we study Liouville type theorem for p-harmonic morphism.

A GENERALIZATION OF LIOUVILLE′S THEOREM ON INTEGRATION IN FINITE TERMS

  • Utsanee, Leerawat;Vichian, Laohakosol
    • Journal of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.13-30
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    • 2002
  • A generalization of Liouville's theorem on integration in finite terms, by enlarging the class of fields to an extension called Ei-Gamma extension is established. This extension includes the $\varepsilon$L-elementary extension of Singer, Saunders and Caviness and contains the Gamma function.

A MONOTONICITY FORMULA AND A LIOUVILLE TYPE THEOREM OF V-HARMONIC MAPS

  • Zhao, Guangwen
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1327-1340
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    • 2019
  • We establish a monotonicity formula of V-harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V-harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ${\pm}holomorphic$ maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V-harmonic maps is considered.

REMARKS ON LIOUVILLE TYPE THEOREMS FOR THE 3D STATIONARY MHD EQUATIONS

  • Li, Zhouyu;Liu, Pan;Niu, Pengcheng
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1151-1164
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    • 2020
  • The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz [14] and Seregin-Wang [18].

PLANCHEREL AND PALEY-WIENER THEOREMS FOR AN INDEX INTEGRAL TRANSFORM

  • Kim, Vu--Tuan;Ali Ismail;Megumi Saigo
    • Journal of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.545-563
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    • 2000
  • An integral transform with the Bessel function Jv(z) in the kernel is considered. The transform is relatd to a singular Sturm-Liouville problem on a half line. This relation yields a Plancherel's theorem for the transform. A Paley-Wiener-type theorem for the transform is also derived.

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Stability and Constant Boundary-Value Problems of f-Harmonic Maps with Potential

  • Kacimi, Bouazza;Cherif, Ahmed Mohammed
    • Kyungpook Mathematical Journal
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    • v.58 no.3
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    • pp.559-571
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    • 2018
  • In this paper, we give some results on the stability of f-harmonic maps with potential from or into spheres and any Riemannian manifold. We study the constant boundary-value problems of such maps defined on a specific Cartan-Hadamard manifolds, and obtain a Liouville-type theorem. It can also be applied to the static Landau-Lifshitz equations. We also prove a Liouville theorem for f-harmonic maps with finite f-energy or slowly divergent f-energy.

On the Generalized of p-harmonic and f-harmonic Maps

  • Remli, Embarka;Cherif, Ahmed Mohammed
    • Kyungpook Mathematical Journal
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    • v.61 no.1
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    • pp.169-179
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    • 2021
  • In this paper, we extend the definition of p-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized p-harmonic maps. We present some new properties for the generalized stress p-energy tensor. We also prove that every generalized p-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.