• East Asian mathematical journal
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• v.39 no.3
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• pp.355-367
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• 2023

Extending [14], we establish the existence of multiple positive solutions for a Schrödinger-type singular elliptic equation: $$\{{-{\Delta}u+V(x)u={\lambda}{\frac{f(u)}{u^{\beta}}},\;x{\in}{\Omega}, \atop u=0,\;x{\in}{\partial}{\Omega},$$ where 0 ∈ Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂Ω, β ∈ [0, 1), f ∈ C[0, ∞), V : Ω → ℝ is a bounded function and λ is a positive parameter. In particular, when f(s) > 0 on [0, σ) and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.7
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• pp.1030-1040
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• 1998

One of the main unresolved issues in large-eddy simulation(LES) of wall-bounded turbulent flows is the requirement of high spatial resolution in the near-wall region, especially in the spanwise direction. Such high resolution required in the near-wall region is generally used throughout the computational domain, making simulations of high Reynolds number, complex-geometry flows prohibitive. A grid-embedding strategy using a nonconforming spectral domain-decomposition method is proposed to address this limitation. This method provides an efficient way of clustering grid points in the near-wall region with spectral accuracy. LES of transitional and turbulent channel flow has been performed to evaluate the proposed grid-embedding technique. The computational domain is divided into three subdomains to resolve the near-wall regions in the spanwise direction. Spectral patching collocation methods are used for the grid-embedding and appropriate conditions are suggested for the interface matching. Results of LES using the grid-embedding strategy are promising compared to LES of global spectral method and direct numerical simulation. Overall, the results show that the spectral domain-decomposition grid-embedding technique provides an efficient method for resolving the near-wall region in LES of complex flows of engineering interest, allowing significant savings in the computational CPU and memory.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.143-159
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• 2013

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.317-332
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• 2005

In this paper we consider the hp-version to solve non-constant coefficients elliptic equations on a bounded, convex polygonal domain ${\Omega}$ in $R^2$. A family $G_p=\{I_m\}$ of numerical quadrature rules satisfying certain properties can be used for calculating the integrals. When the numerical quadrature rules $I_m{\in}G_p$ are used for computing the integrals in the stiffness matrix of the variational form we will give its variational form and derive an error estimate of ${\parallel}u-{\widetilde{u}}^h_p{\parallel}_{1,{\Omega}$.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.821-833
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• 2017

We get a result that shows the existence of infinitely many solutions for a class of the elliptic systems involving subcritical Sobolev exponents nonlinear terms with even functionals on the bounded domain with smooth boundary. We get this result by variational method and critical point theory induced from invariant subspaces and invariant functional.

• The Pure and Applied Mathematics
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• v.5 no.2
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• pp.101-108
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• 1998

For a smoothly bounded n-connected domain $\Omega$ in C, we get a formula representing the relation between the Szego" kernel associated with $\Omega$ and holomorphic mappings obtained from harmonic measure functions. By using it, we show that the coefficient of the above holomorphic map is zero in doubly connected domains.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.619-630
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• 2015

This paper is devoted to investigate the multiple solutions for a class of the cooperative elliptic system involving subcritical Sobolev exponents on the bounded domain with smooth boundary. We first show the uniqueness and the negativity of the solution for the linear system of the problem via the direct calculation. We next use the variational method and the mountain pass theorem in the critical point theory.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.39-44
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• 2005

Let M be a compact hypersurface in hyperbolic space and let A be the area of M and V be the volume of the compact domain bounded by M. In this paper, we find a lower bound for $\frac{A}{V}$ in two cases, M has constant scalar curvature and M has constant mean curvature.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.425-437
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• 2002

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $C^{n}$ and let $z^{0}$ $\in$b$\Omega$ a point of finite type. We also assume that the Levi form of b$\Omega$ is comparable in a neighborhood of $z^{0}$ . Then we get precise estimates of the Bergman kernel function, $K_{\Omega}$(z, w), and its derivatives in a neighborhood of $z^{0}$ . .

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.245-252
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• 2006

In this paper we study the uniform decay estimates of the energy for the nonlinear wave equation of Kirchhoff type $$y'(t)-M({\mid}{\nabla}y(t){\mid}^2){\triangle}y(t)\;+\;{\delta}y'(t)=f(t)$$ with the damping constant ${\delta} > 0$ in a bounded domain ${\Omega}\;{\subset}\;\mathbb{R}^n$.