Dirichlet problem

Japanese: Dirichlet problem（英語表記）Dirichletproblem

… In addition, the problem of finding a solution to Laplace 's equation on a domain D in three-dimensional Euclidean space that satisfies u = φ …(2) or ∂ u /∂ n = φ …(3) on ∂ D , where φ is a function defined on the boundary ∂ D of D, is also a boundary value problem with boundary conditions (2) or (3) (∂ u /∂ n represents the differential coefficient of the solution in the outward normal direction of ∂ D ) . When the boundary condition is given by (2), it is called the Dirichlet problem, and when it is given by (3), it is called the Neumann problem. Problems that find a solution by specifying the asymptotic behavior of the solution as it approaches the boundary infinitesimally are also called boundary value problems. …

From [Harmonic Functions]

…Based on this fact, the theory of two-dimensional harmonic functions is an important chapter in functional theory. Among the boundary value problems related to Laplace's partial differential equation, the following is called the Dirichlet problem. Given a function f on the boundary ∂ D of a domain D , we find a harmonic function u of D that coincides with f . …

From [Potential Theory]

…Gauss's contributions are particularly great. One of the subjects of potential theory is the Dirichlet problem, which is a problem of finding a function that is harmonics in a given domain and coincides with a given function on the boundary. …

*Some of the terminology that refers to the "Dirichlet problem" is listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…　また，三次元ユークリッド空間内の領域D上のラプラス方程式，において，φをDの境界∂D上で定義された関数として，∂D上で，　u＝φ　……(2)　あるいは，　∂u/∂n＝φ　……(3)　を満たす解を求める問題は，やはり(2)，あるいは(3)を境界条件とする境界値問題である(∂u/∂nは∂Dの外向き法線方向への解の微分係数を表す)。境界条件が(2)で与えられているとき，これをディリクレ問題Dirichlet problem，(3)で与えられているとき，ノイマン問題Neumann problemという。　なお，境界に限りなく近づくときの解の漸近的挙動を指定して解を求める問題も，やはり境界値問題と呼ばれる。…