On explicit solvability of an elliptic boundary value problem and its application
On explicit solvability of an elliptic boundary value problem and its application
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椭圆边值问题的显式可解性及其应用
DOI：
10.1080/00036810500137542
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
A. Kozhevnikov
中科院分区：
文献类型：
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作者：
A. Kozhevnikov
在一个任意有界或外部区域Ω（I和Δ是恒等算子和拉普拉斯算子）中为方程（I-Δ）u = f构造一个齐次边界条件，这产生一个具有解u的显式公式的边值问题。这个问题创建了一个同构的适当的Sobolev空间与显式写的逆算子。在本文中，不仅对算子I-Δ，而且对任意偶数阶常系数椭圆型微分算子都得到了所有结果。作为应用，在有界区域或外部区域中齐次方程（I-Δ）u = 0的通常Dirichlet边值问题可化为薄边界层中的积分方程.积分方程的近似解产生一个相当简单的新的数值算法解决二维和三维Dirichlet问题。
A homogeneous boundary condition is constructed for the equation (I − Δ)u = f in an arbitrary bounded or exterior domain Ω ⊆ (I and Δ being the identity operator and the Laplacian), which generates a boundary value problem with an explicit formula of the solution u. The problem creates an isomorphism between the appropriate Sobolev spaces with an explicitly written inverse operator. In the article, all results are obtained not just for the operator I − Δ but also for an arbitrary elliptic differential operator in of an even order with constant coefficients. As an application, the usual Dirichlet boundary value problem for the homogeneous equation (I − Δ)u = 0 in a bounded or exterior domain is reduced to an integral equation in a thin boundary layer. An approximate solution of the integral equation generates a rather simple new numerical algorithm solving the 2D and 3D Dirichlet problem.