A partial differential equation arising in conformal mapping

Garabedian, P. R.

doi:10.2140/pjm.1951.1.485

Cited by 117 publications

(94 citation statements)

References 12 publications

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“…[9,11,30]. In particular, Coffman and Duffin [5] proved that in any two dimensional domain containing a right angle, the positivity preserving property does not hold.…”

Section: Introductionmentioning

confidence: 99%

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Grunau

Robert²

2009

Arch Rational Mech Anal

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Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -equivalently -a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems.It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n , the negative part of the corresponding Green's function is "small" when compared with its singular positive part, provided n ≥ 3.Moreover, the biharmonic Green's function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3.

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“…[9,11,30]. In particular, Coffman and Duffin [5] proved that in any two dimensional domain containing a right angle, the positivity preserving property does not hold.…”

Section: Introductionmentioning

confidence: 99%

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Grunau

Robert²

2009

Arch Rational Mech Anal

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“…[CD,D1,Ga,HJS,KKM,O,Se,ShT]. Even in arbitrarily smooth uniformly convex like mildly eccentric ellipses one has in general change of sign, see [Ga, HJS, ShT].…”

Section: Clamped Plate Modelsmentioning

confidence: 99%

Nonlinear Questions in Clamped Plate Models

Grunau

2009

Milan J. Math.

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Abstract. The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, "linear questions" may be considered as well understood. This changes completely as soon as one poses the simplest "nonlinear question": What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is "no" in general. However, there are many positivity issues as e.g. "almost positivity" to be discussed.Boundary value problems for the "Willmore equation" are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.

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“…We solve this problem in Secs. 2.5 and 2.6 based on results of [11][12][13] (see also [39, §9.4]). In the sections mentioned, an intrinsic isomorphism is established between the subspace ker • A 1 2 and the space H −κ(·) (Γ) = H κ(·) (Γ) * , the dual for the Sobolev space H κ(·) (Γ) with a variable smoothness index κ(s) = 1 − π/α(s).…”

Section: Sapongyan Paradox In Theory Of Platesmentioning

confidence: 99%

“…Indeed, the integral over the narrow curvilinear strip ∂Ω δ \ ∂Ω can be calculated according to the asymptotic decompositions (2.33) and (2.38) and has the left-hand side of (2.46) as the limit. Necessary calculations for the Poisson equation can be found in [40] or in [39, Lemma 11.4.1] and, for general elliptic problems, in [12,13] or [39, §9.4].…”

Section: Proof It Is Easy To Check That Under Conditions (227) ζ(Hmentioning

confidence: 99%

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

Назаров¹,

Sweers

2007

J Math Sci

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Let Ω be a domain with piecewise smooth boundary. In general, it Formulation of the problem and a preliminary description of the results Domain and boundary-value problems.Let Ω ⊂ R 3 be a domain with compact closure Ω and a two-dimensional boundary ∂Ω that is smooth everywhere except for an edge. The edge Γ is assumed to be a smooth, simple, and closed contour in R 3 . At each point s ∈ Γ ⊂ ∂Ω, there exist two planes Π ± (s) that are tangent to the two surfaces ∂Ω ± . These two smooth surfaces remain from the splitting of ∂Ω by Γ. Let α(s) denote the angle between the planes Π + (s) and Π − (s) measured from inside Ω. If(1.1)we say that the domain Ω is convex (locally convex) near the edge; ifthe domain becomes non-convex. If the function α is strictly monotone near a point s 0 ∈ Γ with α(s 0 ) = π, then the edge changes from locally convex to locally concave. In the case α = 2π, we call Γ the front of a crack.In the domain Ω, we consider the boundary-value problem for the biharmonic equation:

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A partial differential equation arising in conformal mapping

Cited by 117 publications

References 12 publications

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Nonlinear Questions in Clamped Plate Models

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

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