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The equivalence of some variational problems for surfaces of prescribed mean curvature

Published online by Cambridge University Press:  17 April 2009

Graham H. Williams
Graham H. Williams
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria.
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Abstract

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One method of finding non-parametric hypersurfaces of prescribed mean curvature which span a given curve in Rn is to find a function which minimizes a particular integral amongst all smooth functions satisfying certain boundary conditions. A new problem can be considered by changing the integral slightly and then minimizing over a larger class of functions. It is possible to show that a solution to this new problem exists under very general conditions and it is usually known as the generalized solution. In this paper we show that the two problems are equivalent in the sense that the least value for the original minimization problem and the generalized problem are the same even though no solution may exist. The case where the surfaces are constrained to lie above an obstacle is also considered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 1 , January 1979 , pp. 87 - 104
Copyright
Copyright © Australian Mathematical Society 1979

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