• Korean Journal of Mathematics
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• 제30권1호
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• pp.121-130
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• 2022

In this paper, we prove that every semi-conformal harmonic map between Riemannian manifolds is L-harmonic map. We also prove a Liouville type theorem for L-harmonic maps.

• 대한수학회보
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• 제56권6호
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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.

• 대한수학회보
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• 제35권2호
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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].

• 대한수학회지
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• 제39권1호
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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.

• 대한수학회논문집
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• 제29권1호
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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.

• 대한수학회보
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• 제56권5호
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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.

• 대한수학회보
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• 제57권5호
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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].

• 대한수학회지
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• 제37권4호
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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.

• 대한수학회지
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• 제54권3호
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• pp.763-772
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• 2017

In this paper, we study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds.

• 대한수학회지
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• 제58권6호
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• pp.1327-1345
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• 2021

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type $$u(x)=\vec{\;l\;}+C_{\ast}{{\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}^n}}}\;{\frac{(1-{\mid}u(y){\mid}^2){\mid}u(y){\mid}^2u(y)-\frac{1}{2}(1-{\mid}u(y){\mid}^2)^2u(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}}dy.$$ Here u : ℝⁿ → ℝ^k is a bounded, uniformly continuous function with k ⩾ 1 and 0 < α < n, $\vec{\;l\;}{\in}\mathbb{R}^k$ is a constant vector, and C_* is a real constant. We prove that ${\mid}\vec{\;l\;}{\mid}{\in}\{0,\frac{\sqrt{3}}{3},1\}$ if u is the finite energy solution. Further, if u is also a differentiable solution, then we give a Liouville type theorem, that is either $u{\rightarrow}\vec{\;l\;}$ with ${\mid}\vec{\;l\;}{\mid}=\frac{\sqrt{3}}{3}$, when |x| → ∞, or $u{\equiv}\vec{\;l\;}$, where ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.