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In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function ''f'' that has values everywhere on the boundary of a region in R''n'', is there a unique continuous function ''u'' twice continuously differentiable in the interior and continuous on the boundary, such that ''u'' is harmonic in the interior and ''u'' = ''f'' on the boundary? This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proved using the maximum principle. ==History== The Dirichlet problem is named after Peter Gustav Lejeune Dirichlet, who proposed a solution by a variational method which became known as Dirichlet's principle. The existence of a unique solution is very plausible by the 'physical argument': any charge distribution on the boundary should, by the laws of electrostatics, determine an electrical potential as solution. However, Karl Weierstrass found a flaw in Dirichlet's argument, and a rigorous proof of existence was found only in 1900 by David Hilbert. It turns out that the existence of a solution depends delicately on the smoothness of the boundary and the prescribed data. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dirichlet problem」の詳細全文を読む スポンサード リンク
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