EXISTENCE THEOREMS OF SOLUTIONS FOR TWO－POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

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An inverse boundary value problem for Maxwell's equations

We consider steady, two-dimensional motions of an incompressible, Newtonian fluid flowing under gravity down an inclined channel. If the bottom of the channel is flat, the flow is the classical Poiseuille-Nusselt flow and the free surface is then a plane parallel to the bottom. Motivated by the recent experimental and numerical studies of Pritchard, Scott & Tavener, we look at bottom configurations which possess some localized, non-uniform structure. We present an existence theory for steady, highly viscous flow over such configurations. An important consequence of our theory is that the steady flows whose existence is established decay exponentially rapidly to the unperturbed Poiseuille-Nusselt flow away from the local variation in the channel bottom profile.     

Some C0-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems

The goal of this work is a systematic presentation of some classes of mixed weak formulations, for general multi-dimensional dipolar gradient elasticity (fourth order) boundary value problems. The displacement field main variable is accompanied by the double stress tensor and the Cauchy stress tensor (case 1 or μ ? τ ? u formulation), the double stress tensor alone (case 2 or μ ? u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or μ ? τ ? κ ? ε ? u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u ? θ ? γ formulation). In all formulations, the respective essential conditions are built in the structure of the solution spaces. For cases 1, 2 and 4, one-dimensional analogues are presented for the purpose of numerical comparison. Moreover, the standard Galerkin formulation is depicted. It is noted that the standard Galerkin weak form demands C¹-continuous conforming basis functions. On the other hand, up to first order derivatives appear in the bilinear forms of the current mixed formulations. Hence, standard C⁰-continuous conforming basis functions may be employed in the finite element approximations. The main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods. In all cases, the associated quadratic energy functionals are formed for the purpose of completeness.     

Multiplicity results for a fourth-order boundary value problem

ＭＵＬＴＩＰＬＩＣＩＴＹＲＥＳＵＬＴＳＦＯＲＡＦＯＵＲＴＨ－ＯＲＤＥＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＭａＲｕｙｕｎ（马如云）ＭａＱｉｎｓｈｅｎｇ（马勤生）（ＲｅｃｅｉｖｅｄＯｃｔ．５．１９９４：ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉｎＺｏｎｇｃｈ...     

Stress-unilateral analysis of discretized cable and membrane structure in the presence of large displacements

Summary The present paper deals with the calculation of structures, some elements of which do not admit compressive stresses. An inequality theory is formulated for the large displacement-calculation of these structures. It is proved that for the numerical calculation of this inequality-problem the algorithms of quadratic minimization must be used.

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Übersicht Im vorliegenden Aufsatz werden Tragwerke behandelt; in denen Elemente vorhanden sind, die keine Druckspannung aufnehmen können. So wird eine Ungleichungstheorie für die Berechnung dieser Tragwerke im Gebiet großer Verschiebungen formuliert. Es wird bewiesen, daß für die numerische Lösung des Ungleichungsproblems die Algorithmen der quadratischen Minimierung benutzt werden müssen.

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The author's grateful thanks are due to the Alexander-von-Humboldt-Stiftung for the financial support of this work at the Institut für Technische Mechanik, RWTH Aachen.     

Convergence of the approximate solution for the elliptic boundary value problem

The convergence of approximate solutions of boundary value problem of fourth order elliptic differential equations with uncontinuous coefficients in a rectangular region is investigated in this paper. This is useful for certain bending problems of rectangular plate on elastic supports.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.     

Contrast structure for singular singularly perturbed boundary value problem

The step-type contrast structure for a singular singularly perturbed problem is shown. By use of the method of boundary function, the formal asymptotic expansion is constructed. At the same time, based on sewing orbit smooth, the existence of the step- type solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of the present results.     

On the solution of nonlinear two-point boundary value problem

In this paper, a non-variational version of a max-min principle is proposed, andan existence and uniqueness result is obtained for the nonlinear two-point boundaryvalue problenl u"+g(t.u)=f(t),u(0)=u(2π)=0     

New results of some existence theorems on nonlinear boundary value problems

IntroductionSeveralpapershavebeenwrittenaboutequationwhereLisalineardifferentialexpression,NacontinuouslyFrechetdifferentialoperatorsuchthatL N'(u)satisfiescertainconditions,andFanoperator,usuallywithboundedrange.AconcreteexampleiswhereLisaseconddifferentialoperator,gsatisfiessomeconditionsrelatingtoL,andfisbounded.AmainlyusedtechniqueisSchauderfixedpointtheorem.Byestablishingapriorbound,itisnaturaltoobtaintheexistCnceofthesolutions.Butthismethodcannotgettheuniquenessofthesolution,noristh…     

Mixed boundary value problem of elasticity for a quarter space

We consider the static elasticity problem for a quarter space with zero displacements on one of its surfaces and with given stresses on the other. The method for solving this problem is based on the use of newunknown functions in the formof a linear combination of the desired displacements, which reduces the system of three Lamé equations to two equations to be solved simultaneously and one equation to be solved separately. The exact solution of this problem was obtained earlier by the same method [1]. But it was shown in [2] that such a solution is exact only under certain restrictions on the given functions. In the present paper, the solution of this problem is constructed without restrictions on the given functions, which necessitates solving a one-dimensional integro-differential equation; this can be done approximately by the orthogonal polynomial method. We present numerical results obtained on the basis of our solution.     

Mixed boundary value problem of heat conduction for infinite slab

Summary An axisymmetric steady state heat conduction boundary value problem having mixed boundary conditions on both faces of an infinite slab, is reduced to a pair of Fredholm integral equations of the second kind. For large values of h, the slab thickness, a solution correct to O(h ^–6) is obtained by expanding the kernels in power series.Presently at Imperial College, London.     

Existence of solutions for fourth-order boundary value problem with parameter

This paper is a sequel to a previous paper (Yang,Y.and Zhang,J.H.Existence of solutions for some fourth-order boundary value problems with parameters.Nonlinear Anal.69(2),1364–1375 (2008)) in which the nontrivial solutions to the fourthorder boundary value problems were studied.In the current work with the same conditions near infinity but different near zero,the positive,negative,and sign-changing solutions are obtained by the critical point theory,retracting property,and invariant sets.     

ON THE EXISTENCE AND STABILITY OF SOLUTION FOR SEMI－HOMOGENEOUS BOUNDARY VALUE PROBLEM

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