• Korean Journal of Mathematics
• /
• v.17 no.4
• /
• pp.419-428
• /
• 2009

We show the existence of at least one nontrivial solution of the homogeneous mixed type nonlinear elliptic problem. Here mixed type nonlinearity means that the nonlinear part contain the jumping nonlinearity and the critical growth nonlinearity. We first investigate the sub-level sets of the corresponding functional in the Soboles space and the linking inequalities of the functional on the sub-level sets. We next investigate that the functional I satisfies the mountain pass geometry in the critical point theory. We obtain the result by the mountain pass method, the critical point theory and variational method.

• Journal of applied mathematics & informatics
• /
• v.15 no.1_2
• /
• pp.237-249
• /
• 2004

This paper deals with existence of positive solutions of nonlinear elliptic singular boundary value problems in a ball. It is shown that results of Grandall et al. [1] and [2] follow as special cases of our results proved in this article.

• Korean Journal of Mathematics
• /
• v.17 no.1
• /
• pp.103-112
• /
• 2009

We show the existence of the solutions for the following nonlinear elliptic problem under the Dirichlet boundary condition. To show the existence of the solutions we use the variational formulation.

• Korean Journal of Mathematics
• /
• v.18 no.3
• /
• pp.323-334
• /
• 2010

We give a theorem for the existence of at least three solutions for the fourth order elliptic boundary value problem with the square growth variable coefficient nonlinear term. We use the variational reduction method and the critical point theory for the associated functional on the finite dimensional subspace to prove our main result. We investigate the shape of the graph of the associated functional on the finite dimensional subspace, (P.S.) condition and the behavior of the associated functional in the neighborhood of the origin on the finite dimensional reduction subspace.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.26 no.4
• /
• pp.244-262
• /
• 2022

In this paper, we analyze the hybridizable discontinuous Galerkin (HDG) method for second-order elliptic equations with nonlinear coefficients, which are used in many fields. We present the HDG method that uses a mixed formulation based on numerical trace and flux. Under assumptions on the nonlinear coefficient and H²-regularity for a dual problem, we prove that the discrete systems are well-posed and the numerical solutions have the optimal order of convergence as a mesh parameter. Also, we provide a matrix formulation that can be calculated using an iterative technique for numerical experiments. Finally, we present representative numerical examples in 2D to verify the validity of the proof of Theorem 3.10.

• Nonlinear Functional Analysis and Applications
• /
• v.28 no.3
• /
• pp.613-622
• /
• 2023

In this paper, finite element method is applied to solve boundary control problem governed by elliptic variational inequality with an infinite number of variables. First, we introduce some important features of the finite element method, boundary control problem governed by elliptic variational inequalities with an infinite number of variables in the case of the control and observation are on the boundary is introduced. We prove the existence of the solution by using the augmented Lagrangian multipliers method. A triangular type finite element method is used.

• Steel and Composite Structures
• /
• v.37 no.2
• /
• pp.175-191
• /
• 2020

In this article, for the first time, the seismic behavior of elliptic-braced moment resisting frame (ELBRF) is assessed through a laboratory program and numerical analyses of FEM specifically focused on the development of global- and local-type failure mechanisms. The ELBRF as a new lateral braced system, when installed in the middle bay of the frames in the facade of a building, not only causes no problem to the opening space of the facade, but also improves the structural behavior. Quantitative and qualitative investigations were pursued to find out how elliptic braces would affect the failure mechanism of ELBRF structures exposed to seismic action as a nonlinear process. To this aim, an experimental test of a ½ scale single-story single-bay ELBRF specimen under cyclic quasi-static loading was run and the results were compared with those for X-bracing, knee-bracing, K-bracing, and diamond-bracing systems in a story base model. Nonlinear FEM analyses were carried out to evaluate failure mechanism, yield order of components, distribution of plasticity, degradation of structural nonlinear stiffness, distribution of internal forces, and energy dissipation capacity. The test results indicated that the yield of elliptic braces would delay the failure mode of adjacent elliptic columns and thus, help tolerate a significant nonlinear deformation to the point of ultimate failure. Symmetrical behavior, high energy absorption, appropriate stiffness, and high ductility in comparison with the conventional systems are some of the advantages of the proposed system.

• Korean Journal of Mathematics
• /
• v.19 no.4
• /
• pp.423-436
• /
• 2011

We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

• Journal of the Korean Mathematical Society
• /
• v.56 no.1
• /
• pp.239-263
• /
• 2019

The $Calder{\acute{o}}n$-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $a({\xi},x_1,x^{\prime})$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables x', uniformly in ${\xi}$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.

• Korean Journal of Mathematics
• /
• v.23 no.4
• /
• pp.619-630
• /
• 2015

This paper is devoted to investigate the multiple solutions for a class of the cooperative elliptic system involving subcritical Sobolev exponents on the bounded domain with smooth boundary. We first show the uniqueness and the negativity of the solution for the linear system of the problem via the direct calculation. We next use the variational method and the mountain pass theorem in the critical point theory.