• Title, Summary, Keyword: Dirichlet space

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S-SHAPED CONNECTED COMPONENT FOR A NONLINEAR DIRICHLET PROBLEM INVOLVING MEAN CURVATURE OPERATOR IN ONE-DIMENSION MINKOWSKI SPACE

  • Ma, Ruyun;Xu, Man
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1891-1908
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    • 2018
  • In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$\{\(\frac{u^{\prime}}{\sqrt{1-u^{{\prime}2}}}\)^{\prime}+{\lambda}a(x)f(u)=0,\;x{\in}(0,1),\\u(0)=u(1)=0$$, where ${\lambda}$ is a positive parameter, $f{\in}C[0,{\infty})$, $a{\in}C[0,1]$. The proofs of main results are based upon the bifurcation techniques.

LERAY-SCHAUDER DEGREE THEORY APPLIED TO THE PERTURBED PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.219-231
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    • 2009
  • We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

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The existence of solutions of a nonlinear wave equation

  • Choi, Q-Heung;Jung, Tack-Sun
    • Communications of the Korean Mathematical Society
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    • v.11 no.1
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    • pp.153-167
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    • 1996
  • In this paper we investigate the existence of solutions of a nonlinear wave equation $u_{tt} - u_{xx} = p(x, t, u)$$ in $H_0$, where $H_0$ is the Hilbert space spanned by eigenfunctions. If p satisfy condition $(p_1) - (p_3)$, this nonlinear gave equation has at least one solution.

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PROPERTIES OF ELASTIC SYMBOLS AND CONSTRUCTION OF SOLUTIONS OF THE DIRICHLET PROBLEM

  • Kawashita, Mishio;Soga, Hideo
    • Communications of the Korean Mathematical Society
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    • v.16 no.3
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    • pp.399-404
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    • 2001
  • We examine plane waves of the elastic reduced wave equation in the half-space, and show that linear combinations of them can cover all plane waves on the boundary. The proof is based on the complex analysis for the symbol in the (dual) variable in the normal direction to the boundary.

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PENALIZED NAVIER-STOKES EQUATIONS WITH INHOMOGENEOUS BOUNDARY CONDITIONS

  • Kim, Hongchul
    • Korean Journal of Mathematics
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    • v.4 no.2
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    • pp.179-193
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    • 1996
  • This paper is concerned with the penalized stationary incompressible Navier-Stokes system with the inhomogeneous Dirichlet boundary condition on the part of the boundary. By taking a generalized velocity space on which the homogeneous essential boundary condition is imposed and corresponding trace space on the boundary, we pose the system to the weak form which the stress force is involved. We show the existence and convergence of the penalized system in the regular branch by extending the div-stability condition.

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Boundary Integral Equation Method by Cubic Spline (Cubic Spline을 사용한 경계요소법)

  • 서승남
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.2 no.1
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    • pp.11-17
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    • 1990
  • Dirichlet boundary value problems originated from unsteady deep water wave propagation are transformed to Boundary Intergral Equation Methods by use of a free surface Green's function and the integral equations are discretized by a cubic spline element method. In order to enhance the stability of the numerical model based on the derived Fredholm integral equation of 1 st kind, the method by Hsiao and MacCamy (1973) is employed. The numerical model is tested against exact solutions for two cases and the model shows very good accuracy.

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UNIQUE POSITIVE SOLUTION FOR A CLASS OF THE SYSTEM OF THE NONLINEAR SUSPENSION BRIDGE EQUATIONS

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.355-362
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    • 2008
  • We prove the existence of a unique positive solution for a class of systems of the following nonlinear suspension bridge equation with Dirichlet boundary conditions and periodic conditions $$\{{u_{tt}+u_{xxxx}+\frac{1}{4}u_{ttxx}+av^+={\phi}_{00}+{\epsilon}_1h_1(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\{v_{tt}+v_{xxxx}+\frac{1}{4}u_{ttxx}+bu^+={\phi}_{00}+{\epsilon}_2h_2(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ where $u^+={\max}\{u,0\},\;{\epsilon}_1,\;{\epsilon}_2$ are small number and $h_1(x,t)$, $h_2(x,t)$ are bounded, ${\pi}$-periodic in t and even in x and t and ${\parallel} h_1{\parallel}={\parallel} h_2{\parallel}=1$. We first show that the system has a positive solution, and then prove the uniqueness by the contraction mapping principle on a Banach space

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STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A RIEMANNIAN MANIFOLD OF NONNEGATIVE RICCI CURVATURE

  • Kim, Jeong-Jin;Yun, Gabjin
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1201-1207
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    • 2013
  • Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of $L^2$ harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.

Bayesian Multiple Comparisons for Normal Variances

  • Kim, Hea-Jung
    • Journal of the Korean Statistical Society
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    • v.29 no.2
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    • pp.155-168
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    • 2000
  • Regarding to multiple comparison problem (MCP) of k normal population variances, we suggest a Bayesian method for calculating posterior probabilities for various hypotheses of equality among population variances. This leads to a simple method for obtaining pairwise comparisons of variances in a statistical experiment with a partition on the parameter space induced by equality and inequality relationships among the variances. The method is derived from the fact that certain features of the hierarchical nonparametric family of Dirichlet process priors, in general, make it amenable to solving the MCP and estimating the posterior probabilities by means of posterior simulation, the Gibbs sampling. Two examples are illustrated for the method. For these examples, the method is straightforward for specifying distributionally and to implement computationally, with output readily adapted for required comparison.

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