• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.2
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• pp.193-207
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• 2014

In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet's principle. We emphasize on the intuition in Riemann's statement on the principle criticized by Weierstrass' requirement of rigor followed by Hilbert's restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion. Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly after Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar$\acute{e}$ and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1139-1151
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• 2022

This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝ^N and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

• Journal for History of Mathematics
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• v.17 no.1
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• pp.43-52
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• 2004

In this paper we investigate the origin of the variational calculus with respect to the extremal principle. We also study the role the extremal principle has played in the development of the calculus of variations. We deal with Dido's isoperimetric problem, Maupertius's least action principle, brachistochrone problem, geodesics, Johann Bernoulli's principle of virtual work, Plateau's minimal surface and Dirichlet principle.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.495-505
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• 2023

In this note, we study a comparison principle for elliptic obstacle problems of p-Laplacian type with L¹-data. As a consequence, we improve some known regularity results for obstacle problems with zero Dirichlet boundary conditions.

• Journal of the Korean Mathematical Society
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• v.36 no.6
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• pp.1033-1046
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• 1999

This paper is devoted to the point wise error estimate up to boundary for the standard finite element solution of Poisson equation with Dirichlet boundary condition. Our new approach used the discrete maximum principle for the discrete harmonic solution. once the mesh in our domain satisfies the $\beta$-condition defined by us, the discrete harmonic solution with dirichlet boundary condition has the discrete maximum principle and the pointwise error should be bounded by L-errors newly obtained.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.197-203
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• 2008

We show the existence of the unique solution of the following system of the nonlinear wave equations with Dirichlet boundary conditions and periodic conditions under some conditions $U_{tt}-U_{xx}+av^+=s{\phi}_{00}+f$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, ${\upsilon}_{tt}-{\upsilon}_{xx}+bu^+=t{\phi}_{00}+g$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, where $u^+$ = max{u, 0}, s, t ${\in}$ R, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ of the wave operator. We first show that the system has a positive solution or a negative solution depending on the sand t, and then prove the uniqueness theorem by the contraction mapping principle on the Banach space.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.715-729
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• 2017

In this paper, we investigate the existence and multiplicity of solutions for systems driven by two non-local integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tools are the Saddle point theorem, Ekeland's variational principle and the Mountain pass theorem.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.155-167
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• 2019

In this article we are concerned with some non-local problems of Kirchhoff type with Dirichlet boundary condition in Orlicz-Sobolev spaces. A result of the existence of infinitely many solutions is established using variational methods and Ricceri's critical points principle modified by Bonanno.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.545-551
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• 2008

We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.457-464
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• 1998

An alternative variational method for a steady tow-dimensional inviscid flow problem to determine the domain given the constant wall velocity and constant vorticity in the whole domain is presented. The uniqueness result is obtained for convex domains.