• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.583-595
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• 2016

In the present note, we deal with $L^2$ harmonic 1-forms on complete submanifolds with weighted $Poincar{\acute{e}}$ inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^2$ harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and $Vit{\acute{o}}rio$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.469-479
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• 2022

In this paper, we obtain some vanishing theorems for p-harmonic 1-forms on locally conformally flat Riemannian manifolds which admit an integral pinching condition on the curvature operators.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.709-723
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• 2022

In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in ℝⁿ⁺¹. When the hypersurface is stable minimal, we show that there is no nontrivial L^2p harmonic 1-form for some p. The our range of p is better than those in [7]. With the same range of p, we also give finiteness results on minimal hypersurfaces with finite index.

• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.625-635
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• 2016

In this paper, we study certain doubly-warped products which admit gradient Ricci solitons with harmonic Weyl curvature and non-constant soliton function. The metric is of the form $g=dx^2_1+p(x_1)^2dx^2_2+h(x_1)^2\;{\tilde{g}}$ on ${\mathbb{R}}^2{\times}N$, where $x_1$, $x_2$ are the local coordinates on ${\mathbb{R}}^2$ and ${\tilde{g}}$ is an Einstein metric on the manifold N. We obtained a full description of all the possible local gradient Ricci solitons.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.15-24
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• 2008

Let $N{\geq}1$ and p > 1. Let ${\Omega}$ be a domain of $\mathbb{R}^N$. In this article we shall establish Kato's inequalities for quasilinear degenerate elliptic operators of the form $A_pu$ = divA(x,$\nabla$u) for $u{\in}K_p({\Omega})$, ), where $K_p({\Omega})$ is an admissible class and $A(x,\xi)\;:\;{\Omega}{\times}\mathbb{R}^N{\rightarrow}\mathbb{R}^N$ is a mapping satisfying some structural conditions. If p = 2 for example, then we have $K_2({\Omega})\;= \;\{u\;{\in}\;L_{loc}^1({\Omega})\;:\;\partial_ju,\;\partial_{j,k}^2u\;{\in}\;L_{loc}^1({\Omega})\;for\;j,k\;=\;1,2,{\cdots},N\}$. Then we shall prove that $A_p{\mid}u{\mid}\;\geq$ (sgn u) $A_pu$ and $A_pu^+\;\geq\;(sgn^+u)^{p-1}\;A_pu$ in D'(${\Omega}$) with $u\;\in\;K_p({\Omega})$. These inequalities are called Kato's inequalities provided that p = 2. The class of operators $A_p$ contains the so-called p-harmonic operators $L_p\;=\;div(\mid{{\nabla}u{\mid}^{p-2}{\nabla}u)$ for $A(x,\xi)={\mid}\xi{\mid}^{p-2}\xi$.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.543-551
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• 1997

We prove a norm inequality of the form $\left\$\mid$ \upsilon \right\$\mid$ \leq (r - 1) \left\$\mid$ u \right\$\mid$_p, 1 < p < \infty$, between a non-negative subharmonic function u and a smooth function $\upsilon$ satisfying $$\mid$\upsilon(0)$\mid$ \leq u(0), $\mid$\nabla\upsilon$\mid$ \leq \nabla u$\mid$$ and $\mid$\Delta\upsilon$\mid$ \leq \alpha\Delta u$, where $\alpha$ is a constant with $0 \leq \alpha \leq 1$. This inequality extends Burkholder's inequality where $\alpha = 1$.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.183-205
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• 2024

In this paper, for an m-dimensional (m ≥ 5) complete non-compact submanifold M immersed in an n-dimensional (n ≥ 6) simply connected Riemannian manifold N with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of M, the norm of its weighted mean curvature vector |H_f| and the weighted real-valued function f, we can obtain: • several one-end theorems for M; • two Liouville theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.

• Journal of the Korean Chemical Society
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• v.59 no.5
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• pp.387-396
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• 2015

The DFT and ab initio calculations have been performed to elucidate hydrogen interaction of HOOO-(H₂O)_n (n=1~5) clusters. The optimized geometries, harmonic vibrational frequencies, and binding energies are predicted at various levels of theory. The trans conformer of HOOO monomer is predicted to be thermodynamically more stable than cis form at the CCSD(T) level of theory. For HOOO-(H₂O)_n clusters, the geometries are optimized at B3LYP/aug-cc-pVTZ and CAM-B3LYP/aug-cc-pVTZ levels of theory. The binding energy of HOOO-H₂O cluster is predicted to be 6.05 kcal/mol at the MP2//CAM-B3LYP/ aug-cc-pVTZ level of theory after zero-point vibrational energy (ZPVE) and basis set superposition error (BSSE) correction. The average binding energy per H₂O is increased according to adding a H₂O moiety in HOOO-(H₂O)_n clusters up to 7.2 kcal/mol for n=5.

• Korean Journal of Head & Neck Oncology
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• v.33 no.2
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• pp.17-22
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• 2017

Background and Objectives: Voice change after thyroidectomy may develop without injury of recurrent laryngeal nerve. Psychogenic or emotional factors related to voice change after thyroidectomy has been rarely studied. In this study, we sought to analyze the impact of anxiety on early state of post-thyroidectomy voice change. Materials and Methods: We made a retrospective chart review of 36 patients who underwent thyroidectomy for papillary thyroid carcinoma and voice exam before surgery, 2 weeks after and 1 month after surgery. All patients included in the study answered a questionnaire for State-Trait Anxiety Inventory ; STAI-KYZ (form Korean YZ). Clinico-pathologic factors and parameters of voice analysis were reviewed to analyze correlation to the anxiety index. Results: No differences were identified between clinicopathologic factors and preoperative parameters of voice analysis between patients with higher and lower level of anxiety. Noise to harmonic ratio (NHR) was higher in those patients with higher level of anxiety, 2 weeks after surgery (p=0.043). However, none of the parameters showed any difference 1 month later. Conclusion: With limited number of patients and short period of follow up, significant impact of preoperative anxiety on postoperative voice change after thyroidectomy could not be identified in this preliminary study.