• East Asian mathematical journal
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• v.39 no.1
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• pp.51-63
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• 2023

We establish the interior L^q regularity estimates for spatial gradient of weak solutions to nonlinear parabolic equations with the inhomogeneity which is given by the divergence and nondivergence data.

• East Asian mathematical journal
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• v.38 no.1
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• pp.129-141
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• 2022

We obtain interior regularity estimates in the weighted Orlicz spaces for viscosity solutions of fully nonlinear uniformly parabolic equations u_t - F(D² u, x, t) = f(x, t) in Q₁ under relaxed structure conditions on the nonlinear operator F.

• 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.

• Communications of the Korean Mathematical Society
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• v.17 no.3
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• pp.495-503
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• 2002

We consider Holder estimates for ∂ in analytic polyhedra. In the case of dimension 2, it preserves exact Ho1der regularity, and it maps bounded (0,1)-forms into BMO with respect to volume measure.

• East Asian mathematical journal
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• v.37 no.5
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• pp.591-598
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• 2021

In this paper we study the Cauchy problem for the Chern-Simons gauged O(3) sigma model. We prove the local existence of solutions with low regularity initial data, observing null forms of the system and applying bilinear estimates for wave-Sobolev space H^{s, b}.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1741-1751
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• 2016

We prove the boundary Harnack principle and the Carleson type estimate for ratios of solutions u/v of non-divergence second order elliptic equations $Lu=a_{ij}D_{ij}+b_iD_iu=0$ in a bounded domain ${\Omega}{\subset}R_n$. We assume that $b_i{\in}L^n({\Omega})$ and ${\Omega}$ is a $H{\ddot{o}}lder$ domain of order ${\alpha}{\in}$ (0, 1) satisfying a strong regularity condition.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.965-978
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• 2014

In this paper we establish sharp $L^p$-regularity estimates for averaging operators with convolution kernel associated to hypersurfaces in $\mathbb{R}^d(d{\geq}2)$ of the form $y{\mapsto}(y,{\gamma}(y))$ where $y{\in}\mathbb{R}^{d-1}$ and ${\gamma}(y)={\sum}^{d-1}_{i=1}{\pm}{\mid}y_i{\mid}^{m_i}$ with $2{\leq}m_1{\leq}{\cdots}{\leq}m_{d-1}$.

• East Asian mathematical journal
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• v.25 no.2
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• pp.221-227
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• 2009

In this paper, we introduce the generalized H$\ddot{o}$lder space with a majorant function and prove the H$\ddot{o}$lder regularity for solutions of the Cauchy-Riemann equation in the generalized Holder spaces on a bounded convex domain in $\mathbb{C}^2$.

• Fisheries and Aquatic Sciences
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• v.21 no.8
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• pp.24.1-24.9
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• 2018

The Korean bottom trawl survey has been deployed on a regular basis for about the last decade as part of groundfish stock assessments. The regularity indicates that they sample groundfish once per grid cell whose sides are half of one latitude and that of one longitude, respectively, and whose inside is furthermore divided into nine nested grids. Unless they have a special reason (e.g., running into a rocky bottom), their sample location is at the center grid of the nine nested grids. Given data collected by the survey, we intended to show how to appropriately estimate not only the survey index of a fish stock but also its uncertainty. For the regularity reason, we applied the systematic sampling theory for the above purposes and compared its results with a reference, which was based on the simple random sampling. When using the survey data about 11 fish stocks, collected by the spring and fall surveys in 2014, the survey indices of those stocks estimated under the systematic sampling were overall more precise than those under the simple random sampling. In estimates of the survey indices in number, the standard errors of those estimates under the systematic sampling were reduced from those under the simple random sampling by 0.23~27.44%, while in estimates of the survey indices in weight, they decreased by 0.04~31.97%. In bias of the estimates, the systematic sampling was the same as the simple random sampling. Our paper is first in formally showing how to apply the systematic sampling theory to the actual data collected by the Korean bottom trawl surveys.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.717-737
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• 2014

We study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic interface problems. Optimal $L^2(L^2)$ and $L^2(H^1)$ error estimates are shown to hold for semidiscrete problem under suitable regularity of the true solution in whole domain. Further, fully discrete scheme based on backward Euler method has also analyzed and optimal $L^2(L^2)$ norm error estimate is established. The error estimates are obtained for fitted finite element discretization based on straight interface triangles.