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Chapter4The Dirichlet problem
Up until now we have always assumed the existence of a function \(u\) satisfying a certain partial differential equation or inequality. In this chapter we now turn to rigorously establishing the existence of solutions to the problem
\begin{equation}
\left\{
\begin{alignedat}{2}
\Delta u \amp= 0 \amp\quad\amp \ina \Omega \\
u \amp= g \amp\quad\amp \ona \partial \Omega
\end{alignedat}
\right.\tag{4.1}
\end{equation}
on bounded open sets \(\Omega \subseteq \R^N\) with smooth enough boundaries. We call (4.1) the Dirichlet problem for Laplace’s equation or simply the Dirichlet problem.
While the Laplacian in (4.1) is simpler than the more general elliptic operators considered in Chapter 3, the arbitrariness of the domain still makes this a highly nontrivial problem. In fact, proofs for more general elliptic problems often have the solvability of the Dirichlet problem for the Laplacian as one of their main ingredients.
In this section we will make heavy use of the following terminology, which should be compared with Definition 3.6.
Definition4.1.
We call a \(C^2\) function \(u\) harmonic if \(\Delta u =
0\text{,}\) subharmonic if \(\Delta u \ge 0\text{,}\) and superharmonic if \(\Delta u\le 0\text{.}\)
