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Nazarov S.A., Sweers G.H. — A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge
Nazarov S.A., Sweers G.H. — A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

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

Авторы: Nazarov S.A., Sweers G.H.

Аннотация:

Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W 2 2 (Ω) of the equation $Δ_x^2u = f$ with the boundary conditions $u = Δ_x^u = 0$ by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ $C^∞(Γ)$ is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case α ∈ (π, 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel and co-kernel of the operators and determining the solution u ∈ $W_2^2$ (Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a nonnegative function f ∈ $L_2$(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property remain open.


Язык: en

Рубрика: Математика/

Тип: Статья

Статус предметного указателя: Неизвестно

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Год издания: 2007

Количество страниц: 25

Добавлена в каталог: 29.06.2009

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