Proceedings of the Centre for Mathematics and its Applications

The Dirichlet problem for the minimal surface equation

Graham Williams

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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.

Article information

Source
Miniconference on nonlinear analysis. Neil S. Trudinger, Graham H. Williams, eds. Proceedings of the Centre for Mathematical Analysis, v. 8. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1985), 233-239

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416336999

Mathematical Reviews number (MathSciNet)
MR799232

Zentralblatt MATH identifier
0569.35036

Citation

Williams, Graham. The Dirichlet problem for the minimal surface equation. Miniconference on nonlinear analysis, 233--239, Centre for Mathematical Analysis, The Australian National University, Canberra AUS, 1984. https://projecteuclid.org/euclid.pcma/1416336999


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