Vol. 70, No. 1, 1977

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Non-continuous dependence of surfaces of least area on the boundary curve

Michael James Beeson

Vol. 70 (1977), No. 1, 11–17

In this paper, we consider the question of continuous dependence associated with the following version of Plateau’s problem: Given a (sufficiently smooth) Jordan curve Γ, find a surface of least area bounded by Γ. In other words, we ask whether a surface SΓ of least area among surfaces bounded by Γ can be found, continuously in Γ. The answer is no; in fact, sometimes one does not even have local continuous dependence. That is, for certain curves Γ0, one cannot find SΓ continuously in Γ, even on any neighborhood of Γ0.

Mathematical Subject Classification 2000
Primary: 53A10
Secondary: 58E15, 49F10
Received: 4 November 1975
Published: 1 May 1977
Michael James Beeson