• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.65-73
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• 2003

In this paper we seek a positive, radially symmetric and energy minimizing solution of an m-Laplacian equation, -div$($\mid${\nabla}u$\mid$^{m-2}$\mid${\nabla}u)\;=\;h(u)$. In the variational sense, the solutions are the critical points of the associated functional called the energy, $J(v)\;=\;\frac{1}{m}\;\int_{R^N}\;$\mid${\nabla}v$\mid$^m\;-\;\int_{R^N}\;H(v)dx,\;where\;H(v)\;=\;{\int_0}^v\;h(t)dt$. A positive, radially symmetric critical point of J can be obtained by solving the constrained minimization problem; minimize{$\int_{R^N}$\mid${\nabla}u$\mid$^mdx$\mid$\;\int_{R^N}\;H(u)d;=\;1$}. Moreover, the solution minimizes J(v).

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.1023-1036
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• 2017

In this paper, we study the existence of positive solutions to the p-Kirchhoff elliptic equation involving general subcritical growth $(a+{\lambda}{\int_{\mathbb{R}^N}{\mid}{\nabla}u{\mid}^pdx+{\lambda}b{\int_{\mathbb{R}^N}{\mid}u{\mid}^pdx)(-{\Delta}_pu+b{\mid}u{\mid}^{p-2}u)=h(u)$, in ${\mathbb{R}}^N$, where a, b > 0, ${\lambda}$ is a parameter and the nonlinearity h(s) satisfies the weaker conditions than the ones in our known literature. We also consider the asymptotics of solutions with respect to the parameter ${\lambda}$.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.333-360
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• 2024

This paper is concerned with the radial solutions of a nonlinear elliptic equation ∆_pu + |x|^𝑙₁ |u|^q₁-1 u + |x|^𝑙₂ |u|^q₂-1 u = 0, x ∈ ℝ^N, where p > 2, N ≥ 1, q₂ > q₁ ≥ 1, -p < 𝑙₂ < 𝑙₁ ≤ 0 and -N < 𝑙₂ < 𝑙₁ ≤ 0. We prove the existence of global solutions, we give their classification and we present the explicit behavior of positive solutions near the origin and infinity.

• 제어로봇시스템학회:학술대회논문집
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• 1998.10a
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• pp.440-444
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• 1998

In this paper, we consider numerical solution of a H-J-B (Hamilton-Jacobi-Bellman) equation of elliptic type arising from the stochastic control problem. For the numerical solution of the equation, we take an approach involving contraction mapping and finite difference approximation. We choose the It(equation omitted) type stochastic differential equation as the dynamic system concerned. The numerical method of solution is validated computationally by using the constructed test case. Map of optimal controls is obtained through the numerical solution process of the equation. We also show how the method applies by taking a simple example of nonlinear spacecraft control.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.31 no.9
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• pp.741-753
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• 2007

The elliptic conceptual second moment model for turbulent heat fluxes, which was proposed on the basis of elliptic-relaxation equation, was applied to calculate the turbulent heat transfer in an axially rotating pipe flow. The model was closely linked to the elliptic blending model which was used for the prediction of Reynolds stress. The effects of rotation on the turbulent characteristics including the mean velocity, the Reynolds stress tensor, the mean temperature and the turbulent heat flux vector were examined by the model. The numerical results by the present model were directly compared to the DNS as well as the experimental results to assess the performance of the model predictions and showed that the behaviors of the turbulent heat transfer in the axially rotating pipe flow were satisfactorily captured by the present models.

• Proceedings of the KSME Conference
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• 2004.04a
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• pp.1601-1606
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• 2004

In this paper, the unsteady behavior of the viscous flow field past an impulsively started elliptic cylinder is studied numerically. In order to analyze flow field, we introduce vortex particle method. The vorticity transport equation is solved by fractional step algorithm which splits into convection term and diffusion term. The convection term is calculated with Biot-Savart law, the no-through boundary condition is employed on solid boundaries. The diffusion term is modified based on the scheme of particle strength exchange. The particle redistributed scheme for general geometry is adapted. The flows around an elliptic cylinder are investigated for various attack angles at Reynolds number 200. The comparison between numerical results of present study and experimental data shows good agreements.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.28 no.7
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• pp.809-817
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• 2004

In this paper, the unsteady behavior of the viscous flow field past an impulsively started elliptic cylinder is studied numerically. In order to analyze flow field, we introduce vortex particle method. The vorticity transport equation is solved by fractional step algorithm which splits into convection term and diffusion term. The convection term is calculated with Biot-Savart law, the no-through boundary condition is employed on solid boundaries. The diffusion term is modified based on the scheme of particle strength exchange. The particle redistributed scheme for general geometry is adapted. The flows around an elliptic cylinder are investigated for various attack angles at Reynolds number 200. The comparison between numerical results of present study and experimental data shows good agreements.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.239-263
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• 2019

The $Calder{\acute{o}}n$-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $a({\xi},x_1,x^{\prime})$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables x', uniformly in ${\xi}$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.1001-1017
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• 2021

In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.387-398
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• 1998

We investigate the existence of radial nodal solutions of the elliptic equation $\Delta$u ＋ h($\mid$x$\mid$)f(u) = 0, in annular domains. It is proved that for each integer k $\geq$ 1, there exist at least one radially symmetric solution which has exactly k nodes.