• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.495-505
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• 2023

In this note, we study a comparison principle for elliptic obstacle problems of p-Laplacian type with L¹-data. As a consequence, we improve some known regularity results for obstacle problems with zero Dirichlet boundary conditions.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1273-1299
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• 2012

We prove the global regularity of weak solutions to a conormal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth conditions on the low order terms. The leading coefficients are in the class of BMO functions with small mean oscillations.

• 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.

• Journal of the Korean Mathematical Society
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• v.37 no.4
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• pp.565-577
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• 2000

We consider regularity questions arising in the degenerate elliptic vector valued variational inequalities -div(｜▽u｜p-2∇u)$\geq$b(x, u, ∇u) with p$\in$(1, $\infty$). It is a generalization of the scalar valued inequalities, i.e., the obstacle problem. We obtain the C1,$\alpha$loc regularity for the solution u under a controllable growth condition of b(x, u, ∇u).

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.385-401
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• 2021

We prove existence and regularity results of solutions for a class of nonlinear singular elliptic problems like $$\{-div\((a(x)+{\mid}u{\mid}^q){\nabla}u\)=\frac{f}{{\mid}u{\mid}^{\gamma}}{\text{ in }}{\Omega},\\{u=0\;on\;{\partial}{\Omega},$$ where Ω is a bounded open subset of ℝℕ(N ≥ 2), a(x) is a measurable nonnegative function, q, �� > 0 and the source f is a nonnegative (not identicaly zero) function belonging to Lm(Ω) for some m ≥ 1. Our results will depend on the summability of f and on the values of q, �� > 0.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.7-15
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• 1995

Multigrid method has been used widely to solve elliptic problems because of its applicability to many class of problems and fast convergence ([1], [3], [9], [10], [11], [12]). The estimate of convergence rate of multigrid is one of the main objectives of the multigrid analysis ([1], [2], [5], [6], [7], [8]). In many problems, the convergence rate depends on the regularity of the solutions([5], [6], [8]). In this paper, we present an improved estimate of reduction factor of multigrid iteration based on the proof in [6].

• Journal of the Korean Mathematical Society
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• v.60 no.3
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• pp.635-682
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• 2023

In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise Hölder continuous but locally Hölder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove the local piecewise gradient Hölder continuity which was obtained by the previous results.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.135-146
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• 1997

For the numerical simulation of wave phenomena either in unbounded domains that it is not feasible to compute solutions on the entire region, it is needed to truncate the original domains to manageable bounded domains whose geometries are simple but usually nonsmooth. On the artificial boundaries thus created, absorbing boundary conditions are taken so that the significant part of waves arriving at the artificial boundaries can be transmitted [5,10,11,16,17,26]$.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1393-1408
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• 2020

In this paper we mainly study existence and regularity of mild solutions to the parabolic-elliptic system of drift-diffusion type with small initial data in Fourier-Besov spaces. To be more detailed, we will explain that global-in-time mild solutions are well-posed and Gevrey regular by means of multilinear singular integrals and Fourier localization argument. Furthermore, we can get time decay rate estimate of mild solutions in Fourier-Besov spaces.