Abstract
In this paper we consider the Dirichlet problem for the minimal surface equation. We assume that $\Omega$ is a bounded open set in $\mathbb{R}^n$ with $C^2$ boundary $\partial \Omega$ and that $\phi$ is a continuous function on $\partial \Omega$ . Then we consider the problem : \[ (P) \hspace{.2in} Find \hspace{.1in} u \in C^2 (\Omega) \bigcap C^0 (\Omega) such \hspace{.1in} that\\ (i) u = \phi \hspace{.1in} on \hspace{.1in} \partial \Omega ,\\ (ii) u \hspace{.1in} satisfies \hspace{.1in} the \hspace{.1in} minimal \hspace{.1in} surface \hspace{.1in} equation \hspace{.1in} in \hspace{.1in} \Omega , \hspace{.1in} that \hspace{.1in} is ,\\ \sum_{i=1}^{n} D_i \left[ \frac{D_{i}u}{1 + |Du|^2} \right] = 0 \hspace{.1in} in \hspace{.1in} \Omega . \] We shall consider two aspects of this problem : firstly, whether or not solutions exist and, secondly, the regularity of solutions.
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