Dirichlet problemDirichletProblem2005-01-16 12:36:492005-06-07 22:12:00Definition% this is the default PlanetMath preamble. as your knowledge
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\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{amsthm}
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
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\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}Suppose $\Omega$ is a domain of $\sR^n$ and $\partial\Omega$ is the boundary of $\Omega$.
Further suppose $f$ is a function $f\colon\partial \Omega\to\sC$. Then the
\emph{Dirichlet problem} is to find a function $\phi\colon \Omega\cup \partial \Omega \to\sC$
such that
\begin{eqnarray*}
\phi &=& f,\quad \text{on $\partial \Omega$}, \\
\nabla^2 \phi &=& 0,\quad \text{in $\Omega$}.
\end{eqnarray*}