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

Journal Article


Abstract


  • 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. © 1979, Australian Mathematical Society. All rights reserved.

Publication Date


  • 1979

Citation


  • Williams, G. H. (1979). The equivalence of some variational problems for surfaces of prescribed mean curvature. Bulletin of the Australian Mathematical Society, 20(1), 87-104. doi:10.1017/S0004972700009114

Scopus Eid


  • 2-s2.0-84973995081

Web Of Science Accession Number


Start Page


  • 87

End Page


  • 104

Volume


  • 20

Issue


  • 1

Abstract


  • 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. © 1979, Australian Mathematical Society. All rights reserved.

Publication Date


  • 1979

Citation


  • Williams, G. H. (1979). The equivalence of some variational problems for surfaces of prescribed mean curvature. Bulletin of the Australian Mathematical Society, 20(1), 87-104. doi:10.1017/S0004972700009114

Scopus Eid


  • 2-s2.0-84973995081

Web Of Science Accession Number


Start Page


  • 87

End Page


  • 104

Volume


  • 20

Issue


  • 1