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By Yehuda Pinchover and Jacob Rubinstein

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Laplace’s equation can be found everywhere. For example, in the heat conduction problems that were introduced earlier, the temperature field is harmonic when temporal equilibrium is achieved. The equation is also fundamental in mechanics, electromagnetism, probability, quantum mechanics, gravity, biology, etc. r The minimal surface equation When we dip a narrow wire in a soap bath, and then lift the wire gently out of the bath, we can observe a thin membrane spanning the wire. The French mathematician Joseph-Louis Lagrange (1736–1813) showed in 1760 that the surface area of the membrane is smaller than the surface area of any other surface that is a small perturbation of it.

He showed that Fermat’s principle implies that his characteristic function must satisfy a certain first-order nonlinear PDE. Hamilton’s characteristic function and characteristic equation are now called the eikonal function and eikonal equation after the Greek word ικων (or ικoν) which means “an image”. Hamilton discovered that the eikonal equation can be solved by integrating it along special curves that he called characteristics. Furthermore, he showed that in a uniform medium, these curves are exactly the straight light rays whose existence has been assumed since ancient times.

27) x(0, s) = s, y(0, s) = 0, u(0, s) = ψ(s). 28) Let us examine the transversality condition: J= 0 s = −s. 29) Thus we expect a unique solution (at least locally) near each point on the initial curve, except, perhaps, the point x = 0. The solution of the characteristic equations is given by (x(t, s), y(t, s), u(t, s)) = ( f 1 (s) cos t + f 2 (s) sin t, f 1 (s) sin t − f 2 (s) cos t, et f 3 (s)). Substituting the initial condition into the solution above leads to the parametric integral surface (x(t, s), y(t, s), u(t, s)) = (s cos t, s sin t, et ψ(s)).

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