2.2. LAPLACE’S EQUATION 21 Substituting into (3), we obtain Laplace’s equation div(Du) = Δu = 0. If u denotes the chemical concentration temperature electrostatic potential, equation (4) is Fick’s law of diffusion Fourier’s law of heat conduction Ohm’s law of electrical conduction. See Feynman–Leighton–Sands [F-L-S, Chapter 12] for a discussion of the ubiquity of Laplace’s equation in mathematical physics. Laplace’s equa- tion arises as well in the study of analytic functions and the probabilistic investigation of Brownian motion. 2.2.1. Fundamental solution. a. Derivation of fundamental solution. One good strategy for inves- tigating any partial differential equation is first to identify some explicit solutions and then, provided the PDE is linear, to assemble more compli- cated solutions out of the specific ones previously noted. Furthermore, in looking for explicit solutions, it is often wise to restrict attention to classes of functions with certain symmetry properties. Since Laplace’s equation is invariant under rotations (Problem 2), it consequently seems advisable to search first for radial solutions, that is, functions of r = |x|. Let us therefore attempt to find a solution u of Laplace’s equation (1) in U = Rn, having the form u(x) = v(r), where r = |x| = (x1 2 + · · · + xn)1/2 2 and v is to be selected (if possible) so that Δu = 0 holds. First note for i = 1,...,n that ∂r ∂xi = 1 2 ( x1 2 + · · · + xn 2 )−1/2 2xi = xi r (x = 0). We thus have uxi = v (r) xi r , uxixi = v (r) xi 2 r2 + v (r) 1 r xi2 r3
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