Proceedings of the Centre for Mathematics and its Applications

The Dirichlet problem for the minimal surface equation

Graham H. Williams

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Abstract

The minimal surface equation is an elliptic equation but it is nonlinear and is not uniformly elliptic. It is the Euler-Lagrange equation for variational problems which involve minimising the area of the graphs of functions. For the most part we will solve the variational problem with Dirichlet boundary values, that is, when the values of the function are prescribed on the boundary of some given set. We will present some existence results using the Direct Method from the Calculus of Variations and also some interior gradient estimates. All of the techniques can be generalised to include more difficult equations but the essence of the ideas is much clearer when dealing with this particular equation especially as it has such strong geometrical meaning. The material presented closely follows Chapters 12 and 13 from the book "Minimal Surfaces and Functions of Bounded Variation" by E. Giusti.

Article information

Source
Instructional Workshop on Analysis and Geometry, Part 1. Tim Cranny and John Hutchinson, eds. Proceedings of the Centre for Mathematics and its Applications, v. 34. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1996), 91-110

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416323471

Mathematical Reviews number (MathSciNet)
MR1394678

Citation

Williams, Graham H. The Dirichlet problem for the minimal surface equation. Instructional Workshop on Analysis and Geometry, Part 1, 91--110, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1996. https://projecteuclid.org/euclid.pcma/1416323471


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