The Laplace Equation

Abstract

The Laplace equation is the archetypal elliptic equation. It appears in many applications when studying the steady state of physical systems that are otherwise governed by hyperbolic or parabolic operators. Correspondingly, elliptic equations require the specification of boundary data only, and the Cauchy (initial-value) problem does not arise.

Exercises

Exercise 11.1

Show that under an orthogonal change of coordinates, the Laplacian retains the form given in Eq. (11.5). If you prefer to do so, work in a two-dimensional setting.

Exercise 11.2

Express the Laplacian in cylindrical coordinates (or polar coordinates, if you prefer to work in two dimensions).

Exercise 11.3

Obtain Eqs. (11.9), (11.10) and (11.11). Show, moreover, that the flux of the gradient of a harmonic function over the boundary of any bounded domain vanishes.

Exercise 11.4

A membrane is extended between two horizontal concentric rigid rings of 500 and 50 mm radii. If the inner ring is displaced vertically upward by an amount of 100 mm, find the resulting shape of the membrane. Hint: Use Eq. (11.23).

Exercise 11.5

Carry out the calculations leading to (11.27). Make sure to use each of the assumptions made about the solution. Find the solution for the two-dimensional case.

Exercise 11.6

Carry out and justify all the steps necessary to obtain Eq. (11.38).

Exercise 11.7

What is the value of G(X, P)? How is this value reconciled with Eq. (11.46)?

Exercise 11.8

Adopting polar coordinate in the plane, with origin at the centre of the circle, and denoting the polar coordinates of X by $\rho , \theta$ and those of Y (at the boundary) by $R, \psi$, show that the solution to Dirichlet’s problem is given by the expression

$$\begin{aligned} u(\rho , \theta ) = \frac{1}{2\pi }\;\int \limits _0^{2\pi }\;\frac{R^2-\rho ^2}{R^2+\rho ^2-2R\;\rho \;\cos (\theta -\psi )}\;h(\psi )\;d\psi . \end{aligned}$$

(11.50)

where $h(\psi )$ represents the boundary data. Use Eqs. (11.48) and (11.49) (with a 2 in the denominator).