A compact imbedding

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort1to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings2are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.

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Existence of Minimizer to the Ginzburg-Landau Free Energy of Ferromagnetic System

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.444-445.717 ◽

2013 ◽

Vol 444-445 ◽

pp. 717-722

Author(s):

Li Mei Li ◽

Hong Luo

Keyword(s):

Free Energy ◽

Sobolev Space ◽

Monotone Operator ◽

Ginzburg Landau ◽

Landau Free Energy ◽

Lower Semi ◽

Ferromagnetic System ◽

Compact Imbedding

In this paper, we obtain the existence of minimizer to Ginzburg-Landau free energy of ferromagnetic system by coercivity and weakly lower semi-continuity of the free energy, where the weakly lower semi-continuity is derived from monotone operator condition and the Sobolev space compact imbedding theorem.

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A compact imbedding result on Lipschitz manifolds

Mathematische Annalen ◽

10.1007/bf01459256 ◽

1991 ◽

Vol 290 (1) ◽

pp. 491-508 ◽

Cited By ~ 2

Author(s):

Kevin McLeod ◽

Rainer Picard

Keyword(s):

Compact Imbedding

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Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations

Abstract and Applied Analysis ◽

10.1155/2013/830715 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17

Author(s):

Hongyong Cui ◽

Jie Xin

Keyword(s):

Phase Space ◽

Initial Data ◽

Wave Equations ◽

Short Wave ◽

Inertial Manifolds ◽

Inertial Manifold ◽

Uniform Attractor ◽

Approximate Inertial Manifolds ◽

Compact Imbedding ◽

Space Method

Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.

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On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters

ISRN Applied Mathematics ◽

10.5402/2011/625908 ◽

2011 ◽

Vol 2011 ◽

pp. 1-33 ◽

Cited By ~ 1

Author(s):

Le Thi Phuong Ngoc ◽

Le Khanh Luan ◽

Tran Minh Thuyet ◽

Nguyen Thanh Long

Keyword(s):

Asymptotic Expansion ◽

Wave Equation ◽

Weak Solution ◽

Nonlinear Wave ◽

Nonlinear Wave Equation ◽

Taylor Expansion ◽

The Other ◽

Small Parameters ◽

Compact Imbedding ◽

Asymptotic Expansion Of Solutions

A Dirichlet problem for a nonlinear wave equation is investigated. Under suitable assumptions, we prove the solvability and the uniqueness of a weak solution of the above problem. On the other hand, a high-order asymptotic expansion of a weak solution in many small parameters is studied. Our approach is based on the Faedo-Galerkin method, the compact imbedding theorems, and the Taylor expansion of a function.

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