Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-02T22:34:24.259Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Gagliardo, Emilio, “Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabilin”, Rend. Sent. Mat. Univ. Padova 27 (1957), 284–305.Google Scholar
[2]Gerhardt, Claus, “Hypersurfaces of prescribed mean curvature over obstacles”, Math. Z. 133 (1973), 169–185.CrossRefGoogle Scholar
[3]Gerhardt, Claus, “Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature”, Math. Z. 139 (1974), 173–198.CrossRefGoogle Scholar
[4]Gerhardt, Claus, “Existence and regularity of capillary surfaces”, Boll. Un. Mat. Ital. (4) 10 (1974), 317–335.Google Scholar
[5]Giaquinta, M., Souček, J., “Esistenza per il problema dell ' area e controesempio di Bernstein”, Boll. Un. Mat. Ital. (4) 9 (1974), 807–817.Google Scholar
[6]Giusti, Enrico, “Superfici cartesiane di area minima”, Rend. Sem. Mat. Fis. Milano 40 (1970), 135–153.CrossRefGoogle Scholar
[7]Giusti, Enrico, Minimal surfaces and functions of bounded variation (Notes on Pure Mathematics, 10. Department of Pure Mathematics, SGS, Department of Mathematics, IAS, Australian National University, Canberra, 1977).Google Scholar
[8]Massari, U., Pepe, L., “Sull ' approssimazione degli aperti lipschitziani di Rn con varietá differenziabili”, Boll. Un. Mat. Ital. (4) 10 (1974), 532–544.Google Scholar
[9]Miranda, Mario, “Distribuzioni aventi derivate misure insiemi di perimetro localmente finito”, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (3) 18 (1964), 27–56.Google Scholar
[10]Miranda, Mario, “Sul minimo dell'integrale del gradiente di una funzione”, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (3) 19 (1965), 626–665.Google Scholar
[11]Santi, Ettore, “Sul problema al contorno per l'quazione delle superfici di area minima su domini limitati qualunque”, Ann. Univ. Ferrara Sez. VII (N.S.) 17 (1972), 13–26.CrossRefGoogle Scholar
[12]Serrin, J., “The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables”, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496.Google Scholar
[13]Simon, Leon, “Boundary regularity of solutions of the non-parametric least area problem”, Ann. of Math. (2) 103 (1976), 429–455.CrossRefGoogle Scholar
[14]Temam, Roger, “Solutions généralisées de certaines équations du type hypersurfaces minima”, Arch. Rat. Mech. Anal. 44 (1971/1972), 121–156.CrossRefGoogle Scholar
[15]Williams, Graham H., “Surfaces of prescribed mean curvature with a volume constraint”, submitted.Google Scholar