![]() ![]() Bulletin of the American Mathematical Society. "The Work Of Jesse Douglas On Minimal Surfaces". ![]() The mean curvature of each component of a soap film is constant. The soap film will assume the shape of a catenoid. Soap films are made of components that are smooth surfaces. Calculus of Variations Sinclair's Soap Film Problem Find the shape of a soap film (i.e., minimal surface) which will fill two inverted conical funnels facing each other is known as Sinclair's soap film problem (Bliss 1925, p. Plateau formulated a set of empirical rules, now known as Plateau’s Laws, for the formation of soap films: 1. Hildebrandt: "Calculus of Variations", Volumes I and II, Springer Verlag Mathematically, the problem falls within the ambit of the calculus of variations. Fomenko: The Plateau Problem (Studies in the Development of Modern Mathematics), ISBN 2-88124-702-4 Struwe: Plateau's Problem and the Calculus of Variations, ISBN 0-2 Rassias, The Problem of Plateau – A tribute to Jesse Douglas and Tibor Rado (River Edge, NJ, 1992). Biography in Encyclopædia Britannica (Aug.That is to say, it is the curve that minimizes the gravitational potential energy. Biography in Dictionary of Scientific Biography (New York 1970–1990) The catenary is the curved configuration y y(x) of a uniform inextensible rope with two fixed endpoints at rest in a constant gravitational field.^ Peter Lax, Mathematician: An Illustrated Memoir, by Reuben Hersh.Other examples may be found in mechanics, electricity, relativity, and thermodynamics. "A new special form of the linear element of a surface". As a physical example, consider the shapes of soap bubbles and raindrops, which are determined by the surface tension and cohesive forces tending to maintain the fixed volume while decreasing the area to a minimum. Proceedings of the National Academy of Sciences. "Solution of the inverse problem of the calculus of variations". "The most general form of the problem of Plateau". "Green's function and the problem of Plateau". ![]() Sophomores (and freshmen with advanced placement) were privileged to get their introduction to real analysis from a Fields medalist. At the time CCNY only offered undergraduate degrees and Professor Douglas taught the advanced calculus course. The American Mathematical Society awarded him the Bôcher Memorial Prize in 1943.ĭouglas later became a full professor at the City College of New York (CCNY), where he taught until his death. Douglas also made significant contributions to the inverse problem of the calculus of variations. The problem, open since 1760 when Lagrange raised it, is part of the calculus of variations and is also known as the soap bubble problem. He was honored for solving, in 1930, the problem of Plateau, which asks whether a minimal surface exists for a given boundary. ĭouglas was one of two winners of the first Fields Medals, awarded in 1936. He then moved to Columbia University as a graduate student, obtaining a PhD in mathematics in 1920. He attended City College of New York as an undergraduate, graduating with honors in Mathematics in 1916. He was born to a Jewish family in New York City, the son of Sarah (née Kommel) and Louis Douglas. 177-194).Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem. Here is a more recent survey by Dierkes: "Singular Minimal Surfaces" (in Geometric Analysis and Nonlinear Partial Differential Equations, Springer (2003), pp. Jost (ed) Calculus of Variations and Geometric Analysis, Int. A short survey of some old and relatively new results concerning well-posedness of (1)-(3) and its multidimensional analogues can be found in the paper by Dierkes and Huisken, "The N-dimensional analogue of the catenary: Prescribed area", in J. Moreover, the corresponding variational problem has no global solutions for all $A\in\mathbb R$. The equilibrium condition for a hanging heavy surface of constant mass density reads Of the property that an inverted catenary supports smooth rides of a square-wheeledĪ model equation for an inextensible, flexible, heavy surface in a gravitational field was deduced by Poisson Lagrange and later the problem was also studied by Poisson (see the references in the linked papers below). This question arose in imagining a higher-dimensional version I have been unsuccessful in finding anything but simulations of solutions Is there a simple analytic description of any of these surfaces,Īnalogous to the $\cosh$ equation for the catenary curve? Which is the surface of revolution formed by a catenary curve. ![]() The middle option above would look something like this when inverted:
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