• The Pure and Applied Mathematics
• /
• v.11 no.4
• /
• pp.309-321
• /
• 2004

For a complete open Riemannian manifold, the ideal boundary consists of points at infinity. The so-called Busemann-functions play the role of distance functions for points at infinity. We study the similarity and difference between Busemann-functions and ordinary distance functions.

• The Pure and Applied Mathematics
• /
• v.25 no.2
• /
• pp.59-71
• /
• 2018

We consider the boundary value problem with a Dirichlet condition for a second order linear uniformly elliptic operator in a non-divergence form. We study some properties of a barrier at infinity which was introduced by Meyers and Serrin to investigate a solution in an exterior domains. Also, we construct a modified barrier for more general domain than an exterior domain.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.103-114
• /
• 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 Mathematical Society
• /
• v.46 no.1
• /
• pp.151-170
• /
• 2009

Let m be a positive integer. We show that for any given real number ${\alpha}\;{\in}\;[0,\;1]$ and complex number $\mu$ with $|\mu|{\leq}1$ which satisfy $e^{2{\pi}i{\alpha}}{\mu}^m\;{\neq}\;1$, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number $\alpha$. Moreover if the given real number $\alpha$ is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.

• Korean Journal of Mathematics
• /
• v.17 no.1
• /
• pp.53-66
• /
• 2009

We show the existence of a nontrivial periodic solution for the superquadratic parabolic equation with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear term at infinity which have continuous derivatives. We use the critical point theory on the real Hilbert space $L_2({\Omega}{\times}(0 2{\pi}))$. We also use the variational linking theorem which is a generalization of the mountain pass theorem.

• Journal of the Korean Mathematical Society
• /
• v.36 no.2
• /
• pp.355-367
• /
• 1999

We prove the existence of 2N-1 distinct ordered positive solutions of a class of semilinear elliptic Dirichlet boundary value problems on Rn when the forcing term has N distinct positive stable zeros and the coefficient function decaying to the zero at infinity.

• Journal of the Korean Mathematical Society
• /
• v.42 no.3
• /
• pp.543-553
• /
• 2005

In this paper, we prove the existence of nonconstant bounded harmonic maps on a Cartan-Hadamard manifold of pinched negative curvature by solving the asymptotic Dirichlet problem. To be precise, given any continuous data f on the boundary at infinity with image within a ball in the normal range, we prove that there exists a unique harmonic map from the manifold into the ball with boundary value f.

• Journal of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.351-361
• /
• 1995

Let M be a complete open Riemannian manifold. When the sectional curvature $K_M$ of M is nonpositive, Gromov has defined, in his lectures [3], the ideal boundary of M, and used it to study the geometric structure of M. In a Hadamard manifold, a simply connected manifold with nonpositive sectional curvature, a point at infinity can be defined as an equivalence class of rays. He proved many interesting theorems using this definition of ideal boundary and the so-called Tit's metric on it. He also suggested a counterpart to this for nonnegative curvature case. This idea has been taken up by Kasue to study the structure of complete open manifolds with asympttically nonnegative curvature [14]. Motivated by these works, we will define an idela boundary of a general noncompact manifold M, and study its structure.

• Proceedings of the Korean Institute of Interior Design Conference
• /
• 2008.05a
• /
• pp.163-168
• /
• 2008

In a today, our concerns have been changed from a single material to relation, communication and time. The human life made up of science, technology, philosophy and art of the modern society can decide whoerness by connecting or replacing a comprehensive and many kinds. This one is exhibited in interior actively. The space of the pluralistic world is sustained by not simple and fixed relation but complex and flexible relation. And it is shown as an undecided phenomenon by effecting and modifying each other. In here, the boundary has a phase of undecided relation but at the same time it can make possibility to be abundant to infinity. The purpose of our study with considering the uncertainty of the boundary is to solve the discord of between division and unify of modern architecture space and suggest new way to be mediation and to occupy independent area.

• Journal of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.287-304
• /
• 2016

In this paper, we consider the problem of existence of conformal metrics with prescribed mean curvature on the unit ball of ${\mathbb{R}}^n$, $n{\geq}3$. Under the assumption that the order of flatness at critical points of prescribed mean curvature function H(x) is ${\beta}{\in}[1,n-2]$, we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula.