• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1211-1223
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• 2016

Let M be an n-dimensional $K{\ddot{a}}hler$ manifold with positive holomorphic bisectional curvature and let ${\Omega}{\Subset}M$ be a pseudoconvex domain of order $n-q$, $1{\leq}q{\leq}n$, with $C^2$ smooth boundary. Then, we study the (weighted) $\bar{\partial}$-equation with support conditions in ${\Omega}$ and the closed range property of ${\bar{\partial}}$ on ${\Omega}$. Applications to the ${\bar{\partial}}$-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ${\ell}_0$ > 0 such that the ${\bar{\partial}}$-Neumann problem and the Bergman projection are regular in the Sobolev space $W^{\ell}({\Omega})$ for ${\ell}$ < ${\ell}_0$.

• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1787-1799
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• 2017

In this paper, we prove basic a priori estimate for the ${\bar{\partial}}$-Neumann problem on an annulus between two pseudoconvex submanifolds of a Stein manifold. As a corollary of the result, we obtain the global regularity for the ${\bar{\partial}}$-problem on the annulus. This is a manifold version of the previous results on pseudoconvex domains.

• Honam Mathematical Journal
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• v.24 no.1
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• pp.67-73
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• 2002

In this paper we discuss on the existence of general solution of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z})+G(z,\;w,\;\bar{w})$ in the Sololev Space $W_{1,p}(D)$, that is generalization of a first order Non-linear Elliptic System of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z}).$

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.409-421
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• 2018

Let X be a complex manifold of dimension n $n{\geqslant}2$ and let ${\Omega}{\Subset}X$ be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and $E^{{\otimes}m}$ is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of $b{\Omega}$, then we solve the ${\bar{\partial}}$-problem with support condition in ${\Omega}$ for forms of type (r, s), $s{\geqslant}q$ with values in $E^{{\otimes}m}$. Moreover, the solvability of the ${\bar{\partial}}_b$-problem on boundaries of weakly q-convex domains with smooth boundary in $K{\ddot{a}}hler$ manifolds are given. Furthermore, we shall establish an extension theorem for the ${\bar{\partial}}_b$-closed forms.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1753-1757
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• 2015

In this note we obtain a unique continuation result for the differential inequality ${\mid}\bar{\partial}u{\mid}{\leq}{\mid}Vu{\mid}$, where $\bar{\partial}=(i{\partial}_y+{\partial}_x)/2$ denotes the Cauchy-Riemann operator and V (x, y) is a function in $L^2(\mathbb{R}^2)$.

• The Journal of Korean Academy of Prosthodontics
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• v.30 no.1
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• pp.25-42
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• 1992

The purpose of this study was to analyze the magnitude and distribution of stress using a photoelastic model from a unilateral distal extention removable partial dentures with five kinds of the direct retainers, that is, the bilaterally designed bar clasp of the cross-arch lingual bar and the unilaterally designed bar clasp, circumferential clasp, mini-Dalbo attachment, and telescope retainer. A photoelastic model for mandible was made of the epoxy resin(PL-1) and hardner (PLH-1) with the acrylic resin teeth used and was coated with plastic cement-1 at the lingual surface of the model, and then five kinds of removable partial dentures were set, A unilateral vertical load of about 16Kg was applied on the first molar and the stress pattern of the photoelastic model under each condition was analyzed by the reflective circular polariscope. The following results were obtained: 1. The conventional removable partial denture with the bilaterally cross arch lingual bar produced the most favorable stress distribution on the residual ridge and supporting structure of abutment teeth than the unilaterally designed removable partial dentures. 2. The unilaterally designed removable partial denture with the bar clasp produced the stress distribution on the residual ridge, except sligtly higher stress concentration on the supporting structure of the abutment teeth, similar to the conventional removable partial denture with the bilaterally designed cross arch lingual bar. 3. On the unilaterally designed removable partial dentures, the bar clasp produced greater stress distribution on the residual ridge and supporting structure of the abutment teeth than the circumferential clasp. 4. On the unilaterally designed removable partial dentures, the mimi-Dalbo attachment produced relatively higher stress concentration on the residual ridge, but produced lesser stress concentration on the supporting structure of the abutment teeth than the other direct retainers. 5. On the unilaterally designed removable partial dentures, the telescope retainer produced uniform stress distribution on the residual ridge, but produced higher stress concentration at the root apex of the terminal abutment tooth than the other direct retainers. 6. On the unilaterally designed removable partial dentures the circumferential clasp and telescope retainer produced slightly higher stress concentration on the residual ridge and supporting structure of the abutment teeth than the bar clasp and mini- Dalbo attachment.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.205-209
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• 2005

In this paper we discuss on the uniquenss of the solution of partial differential equations: (1) $${\frac{{\partial}w}{{\partial}{\bar{z}}}}=F(z,\;w,\;{\frac{{\partial}w}{{\partial}z}}}+G(z,\;w.{\bar{w})$$ in the sobolev space $W_{1,p}(D)$.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.29-42
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• 2018

We construct bounded linear integral operators that giving solutions to the ${\bar{\partial}}$-equation in $L^p$-spaces and with compact supports on a q-convex intersection ($q{\geq}1$) with ${\mathcal{C}}^3$ boundary in $K{\ddot{a}}hler$ manifolds, and we apply it to obtain a Hartogs-like extension theorems for ${\bar{\partial}}$-closed forms for some bidegree.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.241-252
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• 2011

Let $\{\Omega_{\tau}\}_{\tau{\in}I}$ be a family of strictly convex domains in $\mathbb{C}^n$. We obtain explicit estimates for the solution of the $\bar{\partial}$-equation on $\Omega{\times}I$ in H$\ddot{o}$lder space. We also obtain explicit point-wise derivative estimates for the $\bar{\partial}$-equation both in space and parameter variables.

• East Asian mathematical journal
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• v.19 no.1
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• pp.73-80
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• 2003

Let D be a bounded domain in $\mathbb{C}^n$ with $C^2$ boundary. In this paper, we prove the following inequality $${\parallel}u{\parallel}_{p_2,{\alpha}_2}{\lesssim}{\parallel}u{\parallel}_{p_1,{\alpha}_1}+{\parallel}\bar{\partial}u{\parallel}_{p_1,{\alpha}_1+p_1}/2$$, where $1{\leq}p_1{\leq}p_2<\infty,\;{\alpha}_j>0,(n+{\alpha}_1)/p_1=(n+{\alpha}_1)/p_1=(n+{\alpha}_2)/p_2$, and $1/p_2{\geq}1/p_1-1/2n$.