The Goffman-Liu area and Plateau's problem

Authors:
David Bindschadler and Togo Nishiura

Journal:
Proc. Amer. Math. Soc. **82** (1981), 66-70

MSC:
Primary 28A75; Secondary 49F25

DOI:
https://doi.org/10.1090/S0002-9939-1981-0603603-2

MathSciNet review:
603603

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Goffman and Liu defined a lower semicontinuous area for linearly continuous maps from the disk into , showed that is the Lebesgue area when is continuous and thereby extended the notion of area to some discontinuous maps. With the aid of a simple retraction of the punctured disk onto its boundary, a canonical linearly continuous map of the disk is associated with each continuous closed curve defined on the boundary of the disk. When is a Jordan curve, it is shown that the discontinuous map has the property that where is a continuous map of least area spanning from the classical Plateau problem. Finally, the corresponding least area problem in the class of linearly continuous maps is shown to be trivial, that is, the least area is zero in the class of linearly continuous maps.

**[1]**L. Cesari,*Surface area*, Ann. of Math. Studies, no. 35, Princeton Univ. Press, Princeton, N. J., 1956. MR**0074500 (17:596b)****[2]**R. Courant,*Dirichlet principle, conformal mappings, and minimal surfaces*. Interscience, New York, 1950. MR**0036317 (12:90a)****[3]**J. Douglas,*Solution of the problem of Plateau*, Trans. Amer. Math. Soc.**33**(1931), 263-321. MR**1501590****[4]**C. Goffman,*Bounded variation and absolute continuity in the theory of surface area*, Real Anal. Exchange**4**(1978-79), 9-47. MR**525014 (80a:49071)****[5]**C. Goffman and F.-C. Liu,*Discontinuous mappings and surface area*, Proc. London Math. Soc.**20**(1970), 237-248. MR**0254217 (40:7426)****[6]**C. Morrey,*Multiple integrals in the calculus of variations*, Springer-Verlag, New York, 1966. MR**0202511 (34:2380)****[7]**T. Nishiura,*Some examples relating surface area and the area integral*, Real Anal. Exchange**4**(1978-79), 58-65. MR**525015 (81f:28008)****[8]**T. Radó,*On the problem of least area and the problem of Plateau*, Math. Z.**32**(1930), 763-796. MR**1545197**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28A75,
49F25

Retrieve articles in all journals with MSC: 28A75, 49F25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0603603-2

Keywords:
Lebesgue area,
Fréchet curve,
Plateau problem,
minimal area

Article copyright:
© Copyright 1981
American Mathematical Society