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The Planar Laplace Equation

The fundamental partial differential equations that govern the equilibrium mechanics of multi-dimensional media are the Laplace equation and its inhomogeneous counterpart, the Poisson equation. The Laplace equation is arguably the most important differential equation in all of applied mathematics. It arises in an astonishing variety of mathematical and physical systems, ranging through fluid mechanics, electromagnetism, potential theory, solid mechanics, heat conduction, geometry, probability, number theory, and on and on. The solutions to the Laplace equation are known as “harmonic functions”, and the discovery of their many remarkable properties forms one of the most significant chapters in the history of mathematics. In this chapter, we concentrate on the Laplace and Poisson equations in a two-dimensional (planar) domain. Their status as equilibrium equations implies that the solutions are determined by their values on the boundary of the domain. As in the one-dimensional equilibrium boundary value problems, the principal cases are Dirichlet or fixed, Neumann or free, and mixed boundary conditions arise. In the introductory section, we shall briefly survey the basic boundary value problems associated with the Laplace and Poisson equations. We also take the opportunity to summarize the crucially important tripartite classification of planar second order partial differential equations: elliptic, such as the Laplace equation; parabolic, such as the heat equation; and hyperbolic, such as the wave equation. Each species has quite distinct properties, both analytical and numerical, and each forms an essentially distinct discipline. Thus, by the conclusion of this chapter, you will have encountered all three of the most important genres of partial differential equations. The most important general purpose method for constructing explicit solutions of linear partial differential equations is the method of separation of variables. The method will be applied to the Laplace and Poisson equations in the two most important coordinate systems — rectangular and polar. Linearity implies that we may combine the separable solutions, and the resulting infinite series expressions will play a similar role as for the heat and wave equations. In the polar coordinate case, we can, in fact, sum the infinite series in closed form, leading to the explicit Poisson integral formula for the solution. More sophisticated techniques, relying on complex analysis, but (unfortunately) only applicable to the two-dimensional case, will be deferred until Chapter 16. Green’s formula allows us to properly formulate the Laplace and Poisson equations in self-adjoint, positive definite form, and thereby characterize the solutions via a minimization principle, first proposed by the nineteenth century mathematician Lejeune Dirichlet, who also played a crucial role in putting Fourier analysis on a rigorous foundation. Minimization forms the basis of the most important numerical solution technique — the finite