• Title, Summary, Keyword: unbounded domain

### ON SOME UNBOUNDED DOMAINS FOR A MAXIMUM PRINCIPLE

• CHO, SUNGWON
• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.13-19
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• 2016
• In this paper, we study some characterizations of unbounded domains. Among these, so-called G-domain is introduced by Cabre for the Aleksandrov-Bakelman-Pucci maximum principle of second order linear elliptic operator in a non-divergence form. This domain is generalized to wG-domain by Vitolo for the maximum principle of an unbounded domain, which contains G-domain. We study the properties of these domains and compare some other characterizations. We prove that sA-domain is wG-domain, but using the Cantor set, we are able to construct a example which is wG-domain but not sA-domain.

### Time-domain analyses of the layered soil by the modified scaled boundary finite element method

• Lu, Shan;Liu, Jun;Lin, Gao;Wang, Wenyuan
• Structural Engineering and Mechanics
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• v.55 no.5
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• pp.1055-1086
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• 2015
• The dynamic response of two-dimensional unbounded domain on the rigid bedrock in the time domain is numerically obtained. It is realized by the modified scaled boundary finite element method (SBFEM) in which the original scaling center is replaced by a scaling line. The formulation bases on expanding dynamic stiffness by using the continued fraction approach. The solution converges rapidly over the whole time range along with the order of the continued fraction increases. In addition, the method is suitable for large scale systems. The numerical method is employed which is a combination of the time domain SBFEM for far field and the finite element method used for near field. By using the continued fraction solution and introducing auxiliary variables, the equation of motion of unbounded domain is built. Applying the spectral shifting technique, the virtual modes of motion equation are eliminated. Standard procedure in structural dynamic is directly applicable for time domain problem. Since the coefficient matrixes of equation are banded and symmetric, the equation can be solved efficiently by using the direct time domain integration method. Numerical examples demonstrate the increased robustness, accuracy and superiority of the proposed method. The suitability of proposed method for time domain simulations of complex systems is also demonstrated.

### CONTROLLABILITY OF SECOND ORDER SEMI-LINEAR NEUTRAL IMPULSIVE DIFFERENTIAL INCLUSIONS ON UNBOUNDED DOMAIN WITH INFINITE DELAY IN BANACH SPACES

• Chalishajar, Dimplekumar N.;Acharya, Falguni S.
• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.813-838
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• 2011
• In this paper, we prove sufficient conditions for controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces using the theory of strongly continuous Cosine families. We shall rely on a fixed point theorem due to Ma for multi-valued maps. The controllability results in infinite dimensional space has been proved without compactness on the family of Cosine operators.

### ON A NEUMANN PROBLEM AT RESONANCE FOR NONUNIFORMLY SEMILINEAR ELLIPTIC SYSTEMS IN AN UNBOUNDED DOMAIN WITH NONLINEAR BOUNDARY CONDITION

• Hoang, Quoc Toan;Bui, Quoc Hung
• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1669-1687
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• 2014
• We consider a nonuniformly nonlinear elliptic systems with resonance part and nonlinear Neumann boundary condition on an unbounded domain. Our arguments are based on the minimum principle and rely on a generalization of the Landesman-Lazer type condition.

### ON UNBOUNDED SUBNOMAL OPERATORS

• Jin, Kyung-Hee
• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.65-70
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• 1993
• In this paper we will extend some notions of bounded linear operators to some unbounded linear operators. Let H be a complex separable Hilbert space and let B(H) denote the algebra of bounded linear operators. A closed densely defind linear operator S in H, with domain domS, is called subnormal if there is a Hilbert space K containing H and a normal operator N in K(i.e., $N^{*}$N=N $N^*/)such that domS .subeq. domN and Sf=Nf for f .mem. domS. we will show that the Radjavi and Rosenthal theorem holds for some unbounded subnormal operators; if$S_{1}$and$S_{2}$are unbounded subnormal operators on H with dom$S_{1}$= dom$S^{*}$$_{1} and dom S_{2}=dom S^{*}$$_{2}$and A .mem. B(H) is injective, has dense range and$S_{1}$A .coneq. A$S^{*}$$_{2}, then S_{1} and S_{2} are normal and S_{1}.iden. S^{*}$$_{2}$.2}$.X>.

### Dyadic Greens Function for an Unbounded Anisotropic Medium in Cylindrical Coordinates

• Kai Li;Park, Seong-Ook;Pan, Wei-Yan
• Journal of electromagnetic engineering and science
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• v.1 no.1
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• pp.54-59
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• 2001
• The dyadic Greens function for an unbounded anisotropic medium is treated analytically in the Fourier domain. The Greens function, which is expressed as a triple Fourier integral, can be next reduced to a double integral by performing the integral, by performing the integration over the longitudinal Fourier variable or the transverse Fourier variable. The singular behavior of Greens is discussed for the general anisotropic case.

### THE GLOBAL ATTRACTOR OF THE 2D G-NAVIER-STOKES EQUATIONS ON SOME UNBOUNDED DOMAINS

• Kwean, Hyuk-Jin;Roh, Jai-Ok
• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.731-749
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• 2005
• In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

### ON EXISTENCE OF WEAK SOLUTIONS OF NEUMANN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING p-LAPLACIAN IN AN UNBOUNDED DOMAIN

• Hang, Trinh Thi Minh;Toan, Hoang Quoc
• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1169-1182
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• 2011
• In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+b(x){\mid}u{\mid}^{p-2}u=f(x,\;u),\;p{\geq}2$$ in an unbounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, with sufficiently smooth bounded boundary ${\partial}{\Omega}$, where $h(x){\in}L_{loc}^1(\overline{\Omega})$, $\overline{\Omega}={\Omega}{\cup}{\partial}{\Omega}$, $h(x){\geq}1$ for all $x{\in}{\Omega}$. The proof of main results rely essentially on the arguments of variational method.

### SOLVABILITY FOR SECOND-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON AN UNBOUNDED DOMAIN AT RESONANCE

• Yang, Ai-Jun;Wang, Lisheng;Ge, Weigao
• The Pure and Applied Mathematics
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• v.17 no.1
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• pp.39-49
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• 2010
• This paper deals with the second-order differential equation (p(t)x'(t))' + g(t)f(t, x(t), x'(t)) = 0, a.e. in (0, $\infty$) with the boundary conditions $$x(0)={\int}^{\infty}_0g(s)x(s)ds,\;{lim}\limits_{t{\rightarrow}{\infty}}p(t)x'(t)=0,$$ where $g\;{\in}\;L^1[0,{\infty})$ with g(t) > 0 on [0, $\infty$) and ${\int}^{\infty}_0g(s)ds\;=\;1$, f is a g-Carath$\acute{e}$odory function. By applying the coincidence degree theory, the existence of at least one solution is obtained.

### ON SOME PROPERTIES OF BARRIERS AT INFINITY FOR SECOND ORDER UNIFORMLY ELLIPTIC OPERATORS

• Cho, Sungwon
• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.59-71
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• 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.