A Singular Nonlinear Boundary Value Problem: Membrane Response of a Spherical Cap

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the formwhere G is a compact set in R n . Here α[u] is a positive functional and f is positive, while k and γ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

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“…A special case of the BC (1.3) also occurs in studying the membrane response of a spherical cap [1,3,17,18].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Non-Local Boundary-Value Problems and Perturbed Hammerstein Integral Equations

Infante¹,

Webb²

2006

Proceedings of the Edinburgh Mathematical Society

111

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“…In addition, if 7 = 2 and C is small, (4) approximates the shape of a section of a sphere, as Dickey [7] showed. This spherical-cap case with v = 0 is a well-known problem, which has been studied by Reissner [15], Baxley [1], and Goldberg [10].…”

Section: Introductionmentioning

confidence: 99%

“…Callegari and Reiss [3]; Callegari, Keller, and Reiss [4]; Dickey [8]; Weinitschke [16]; and Hencky [11] considered the plane membrane (C -0); in [3] Callegari and Reiss showed that for the plane membrane, the stress problem always has a solution for S > 0, as does the displacement problem for r = 0 (the fixed-boundary case). Baxley [1] proved that for the spherical cap also, the displacement problem with T -0 has a solution.…”

Section: Introductionmentioning

confidence: 99%

Circularly symmetric deformation of shallow elastic membrane caps

Johnson¹

1997

Quart. Appl. Math.

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Abstract.We consider shallow elastic membrane caps that are rotationally symmetric in their undeformed state, and investigate their deformation under small uniform vertical pressure and a given boundary stress or boundary displacement.To do this we use the small-strain theory developed by Bromberg and Stoker, Reissner, and Dickey We deal with the two-parameter family of membranes whose undeformed configuration is given in cylindrical coordinates aswhich includes the spherical cap as a special case (7 = 2 and C small). We show that if 7 > 4/3 then a circularly symmetric deformation is possible for any positive boundary stress (or any boundary displacement) and any positive pressure, but if 1 < 7 < 4/3 then no circularly symmetric deformation is possible if the stress and pressure are positive and small (or for non-positive boundary displacement and small positive pressure).1. Introduction. This study addresses how an elastic membrane cap deforms under the influence of particular stress and body force. We show, among other things, that under certain conditions a cap that is initially rotationally symmetric will fail to have a symmetric configuration after deformation.To be specific, we consider an elastic membrane cap that is shallow (that is, nearly flat) and rotationally symmetric in its undeformed state (or reference configuration), and then investigate the shape that the cap takes on when radial stress is applied on the boundary and a small uniform vertical pressure P is applied to the membrane. We use the small-strain, small-pressure theory developed by Bromberg and Stoker [2], Reissner [15], and Dickey [5,6] (a generalization to curved membranes of the Foppl theory for plane membranes), which allows for large displacements of the membrane.Dickey [5,6,7] showed that under the assumptions of small strain and small pressure, the radial stress <7r on a membrane whose undeformed profile is given in cylindrical coordinates by

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“…Problem (1.1) arises in the transport of coal slurries down conveyor belts [19], boundary layer theory [10] and chemical reactor theory [6,7]. Due to the important applications, problem (1.1) has received considerable attention during the last few decades.…”

Section: Introductionmentioning

confidence: 99%