数学中，狄利克雷问题（Dirichlet problem）是寻找一个函数，使其为给定区域内一个指定的偏微分方程（PDE）的解，且在边界上取预定值。

对许多偏微分方程，狄利克雷问题都可解，但最初是对拉普拉斯方程提出来的。在这种情形下问题可如下表述：

这个条件称为狄利克雷边界条件。最主要的问题是证明解的存在性，因惟一性可利用极大值原理（Maximum principle）证明。

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: