• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1143-1149
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• 2020

Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces are established.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.257-266
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• 1999

We used Carleson and Chang's method to give another proof of the Moser-Trudinger inequality which was known as a limiting case of the Sobolev imbedding theorem and at the same time we get sharper information for the bound.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.3
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• pp.527-529
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• 2012

It is proved that the fractional Sobolev spaces $W^s_p(\mathbb{R}^n)$ 0 < $s$ < $n$, are compactly embedded into Lebesgue spaces $L^q(\Omega)$ for some bounded set $\Omega$­.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.73-95
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• 1999

We prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincar inequality and the finite covering condition at infinity on each end, then every positive harmonic function on the manifold is asymptotically constant at infinity on each end. This result is a direct generalization of those of Yau and of Li and Tam.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.103-114
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• 1997

The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.2
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• pp.67-77
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• 2007

This paper is concerned with the proof of so-called Bramble-Hilbert Lemma. We present that $Poincar{\acute{e}}'s$ inequality in [3] implies one of results of Morrey which is crucial in the proof. In this point of view, we recognize that removing the average term in $Poincar{\acute{e}}'s$ inequality fulfills a crucial role in the proof of Bramble-Hilbert Lemma. It is accomplished by adding some polynomial of degree one less than the degree of the Sobolev space in the outset. So, the condition annihilating the set of polynomials $P_{k-1}$ of degree k - 1 is required necessarily in Bramble-Hilbert Lemma.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.261-269
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• 2008

Let $B_1$ be a unit ball in $R^n(n{\geq}3)$, and $2^*=2n/(n-2)$ be the critical Sobolev exponent for the embedding $H_0^1(B_1){\hookrightarrow}L^{2^*}(B_1)$. By using a variant of Pohoz$\check{a}$aev's identity, we prove the nonexistence of nodal solutions for the Dirichlet problem $-{\Delta}u-{\mu}\frac{u}{{\mid}x{\mid}^2}={\lambda}u+{\mid}u{\mid}^{2^*-2}u$ in $B_1$, u=0 on ${\partial}B_1$ for suitable positive numbers ${\mu}$ and ${\nu}$.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.263-271
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• 2006

In this paper, a quasilinear second-order system with periodic boundary conditions is studied. By the least action principle and classical theorems of variational calculus, existence results of periodic solutions are obtained.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.823-855
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• 1996

We study Dirichlet forms and the associated diffusion processes for the Gibbs measures related to the quantum unbounded spin systems (lattice boson systems) interacting via superstable and regular potentials. This work is a continuation of the author's previous study on the classical systems [LPY] to the quantum cases. In [LPY], we constructed Dirichlet forms and the associated diffusion processes for the Gibbs measures of classical unbounded spin systems. Furthermore, we also showed the essential self-adjointness of the Dirichlet operator and the log-Sobolev inequality for any Gibbs measure under appropriate conditions on the potentials. In this atudy we try to extend the results of the classical systems to the quantum cases. Because of some technical difficulties, we are only able to construct a Dirichlet form and the associated diffusion process for any Gibbs measure of the quantum systems. We utilize the general scheme of the previous work on the theory in infinite dimensional spaces [AH-K1-2, AKR, AR1-2, Kus, MR, Ro, Sch] and the ideas we employed in our study of the calssical systems ]LPY].