• Title/Summary/Keyword: Dirichlet Condition

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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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BIFURCATION PROBLEM FOR THE SUPERLINEAR ELLIPTIC OPERATOR

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.20 no.3
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    • pp.333-341
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    • 2012
  • We investigate the number of solutions for the superlinear elliptic bifurcation problem with Dirichlet boundary condition. We get a theorem which shows the existence of at least $k$ weak solutions for the superlinear elliptic bifurcation problem with boundary value condition. We obtain this result by using the critical point theory induced from invariant linear subspace and the invariant functional.

NOTES ON THE SPACE OF DIRICHLET TYPE AND WEIGHTED BESOV SPACE

  • Choi, Ki Seong
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.2
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    • pp.393-402
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    • 2013
  • For 0 < $p$ < ${\infty}$, ${\alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $\mathfrak{D}^p_{p-1}$, then ${\int}_{1}^{0}M_{p}^{p}(r,f^{\prime})(1-r)^{p-1}rdr$ < ${\infty}$ and ${\int}_{1}^{0}M_{(2+{\alpha})p}^{(2+{\alpha})p}(r,f^{\prime})(1-r)^{(2+{\alpha})p+{\alpha}}rdr$ < ${\infty}$ where $M_p(r,f)=\[\frac{1}{2{\pi}}{\int}_{0}^{2{\pi}}{\mid}f(re^{it}){\mid}^pdt\]^{1/p}$. For 1 < $p$ < $q$ < ${\infty}$ and ${\alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathfrak{D}^p_{\alpha}}$ for all $f{\in}\mathfrak{D}^p_{\alpha}$, then ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathcal{B}_p(q)}$ where $\mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${\mu}$ such that ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.

Identifying differentially expressed genes using the Polya urn scheme

  • Saraiva, Erlandson Ferreira;Suzuki, Adriano Kamimura;Milan, Luis Aparecido
    • Communications for Statistical Applications and Methods
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    • v.24 no.6
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    • pp.627-640
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    • 2017
  • A common interest in gene expression data analysis is to identify genes that present significant changes in expression levels among biological experimental conditions. In this paper, we develop a Bayesian approach to make a gene-by-gene comparison in the case with a control and more than one treatment experimental condition. The proposed approach is within a Bayesian framework with a Dirichlet process prior. The comparison procedure is based on a model selection procedure developed using the discreteness of the Dirichlet process and its representation via Polya urn scheme. The posterior probabilities for models considered are calculated using a Gibbs sampling algorithm. A numerical simulation study is conducted to understand and compare the performance of the proposed method in relation to usual methods based on analysis of variance (ANOVA) followed by a Tukey test. The comparison among methods is made in terms of a true positive rate and false discovery rate. We find that proposed method outperforms the other methods based on ANOVA followed by a Tukey test. We also apply the methodologies to a publicly available data set on Plasmodium falciparum protein.

DOUBLY NONLINEAR PARABOLIC EQUATIONS RELATED TO THE LERAY-LIONS OPERATORS: TIME-DISCRETIZATION

  • Shin, Ki-Yeon;Kang, Su-Jin
    • East Asian mathematical journal
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    • v.26 no.3
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    • pp.403-413
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    • 2010
  • In this paper, we consider a doubly nonlinear parabolic equation related to the Leray-Lions operator with Dirichlet boundary condition and initial data given. By exploiting a suitable implicit time-discretization technique, we obtain the existence of global strong solution.

BOUNDARY VALUE PROBLEM FOR A CLASS OF THE SYSTEMS OF THE NONLINEAR ELLIPTIC EQUATIONS

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.1
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    • pp.67-76
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    • 2009
  • We show the existence of at least two nontrivial solutions for a class of the systems of the nonlinear elliptic equations with Dirichlet boundary condition under some conditions for the nonlinear term. We obtain this result by using the variational linking theory in the critical point theory.

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NONLINEAR BIHARMONIC PROBLEM WITH VARIABLE COEFFICIENT EXPONENTIAL GROWTH TERM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.18 no.3
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    • pp.277-288
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    • 2010
  • We consider the nonlinear biharmonic equation with coefficient exponential growth term and Dirichlet boundary condition. We show that the nonlinear equation has at least one bounded solution under the suitable conditions. We obtain this result by the variational method, generalized mountain pass theorem and the critical point theory of the associated functional.

NONLINEAR BIHARMONIC EQUATION WITH POLYNOMIAL GROWTH NONLINEAR TERM

  • JUNG, TACKSUN;CHOI, Q-HEUNG
    • Korean Journal of Mathematics
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    • v.23 no.3
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    • pp.379-391
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    • 2015
  • We investigate the existence of solutions of the nonlinear biharmonic equation with variable coefficient polynomial growth nonlinear term and Dirichlet boundary condition. We get a theorem which shows that there exists a bounded solution and a large norm solution depending on the variable coefficient. We obtain this result by variational method, generalized mountain pass geometry and critical point theory.