Plateau’s Problem

Richard Courant
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Richard Courant: New York University, Institute for Mathematics and Mechanics

Chapter Chapter III in Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, 1950, pp 95-139 from Springer

Abstract: Abstract Intimately connected with Dirichlet’s Principle and conformal mapping is Plateau’s problem which has long challenged mathematicians by the contrast between its simplicity of statement and difficulty of solution: to find the surface G of least area spanned in a given closed Jordan curve γ. If the surface G is represented in x,y,z-space by a function z(x,y) with continuous derivatives, the area A is given by $$ A = {\rm{ }}{(1 + z_x^2 + z_y^2)^{1/2}}dxdy, $$ where B is the domain in the x,y-plane bounded by the projection β of γ. The surface G is obtained as solution of the boundary value problem for Euler’s (non-linear) differential equation $$ {z_{xx}}(1 + z_y^2) - 2{z_{xy}}{z_x}{z_y} + {z_{yy}}(1 + z_x^2) = 0 $$

Keywords: Unit Circle; Minimal Surface; Conformal Mapping; Jordan Curve; Isoperimetric Inequality (search for similar items in EconPapers)
Date: 1950
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DOI: 10.1007/978-1-4612-9917-2_4

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