16 1. Introduction Figure 1.9. Catenoid integral of the form S = Ω 1 + fx 2 + fy 2 dx dy (1.46) (see Exercise 1.6.7), where ∂Ω is the projection of the closed curve C onto the (x, y) plane and Ω is the interior of this projection. This problem has been known, starting with Lebesgue (1902), as Plateau’s problem, in honor of Joseph Plateau’s extensive experiments (Plateau, 1873) with soap films. Lagrange showed that a surface that minimizes integral (1.46) must satisfy the minimal surface equation (1 + f 2 y ) fxx − 2 fx fy fxy + (1 + f 2 x ) fyy = 0 , (1.47) a quasilinear, elliptic, second-order, partial differential equation. Dif- ferent constraints on the function f(x, y) (e.g., Exercise 1.6.10) yield different minimal surfaces.

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