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
Canonical name	DirichletProblem
Date of creation	2013-03-22 14:57:06
Last modified on	2013-03-22 14:57:06
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Definition
Classification	msc 31B05
Classification	msc 31A05
Classification	msc 31B15
Related topic	HarmonicFunction