# Dirichlet problem

Suppose $\Omega$ is a domain of $\mathbb{R}^{n}$ and $\partial\Omega$ is the boundary of $\Omega$. Further suppose $f$ is a function $f\colon\partial\Omega\to\mathbb{C}$. Then the Dirichlet problem is to find a function $\phi\colon\Omega\cup\partial\Omega\to\mathbb{C}$ such that

 $\displaystyle\phi$ $\displaystyle=$ $\displaystyle f,\quad\text{on \partial\Omega},$ $\displaystyle\nabla^{2}\phi$ $\displaystyle=$ $\displaystyle 0,\quad\text{in \Omega}.$
Title Dirichlet problem DirichletProblem 2013-03-22 14:57:06 2013-03-22 14:57:06 matte (1858) matte (1858) 7 matte (1858) Definition msc 31B05 msc 31A05 msc 31B15 HarmonicFunction