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A solution to Laplace’s equation is a function u satisfying ∆u = 0.
When combined with boundary conditions such as

u = g on ∂U,

finding a solution to Laplace’s equation is referred to as the Dirichlet Problem.
Laplace’s equation in 2 and 3 dimensions occurs in time-independent problems involv-
ing potentials (e.g. electrostatic, gravitational) and velocity in fluid mechanics. A solution
to Laplace’s equation can also be interpreted as a steady-state temperature distribution.
In a typical interpretation, u denotes the density or concentration of some quantity in
equilibrium:
∆u = 0 – law of diffusion (u denotes chemical concentration)
∆u = 0 – law of heat conduction (u denotes temperature)
∆u = 0 – law of electrical conduction (u denotes electrostatic potential).
The equation we are going to solve is Laplace’s equation in two variables (x and y),
which can be written as
∆u = u xx + u yy = 0 (10.36)
It can be seen that equation (10.36) is an elliptic partial differential equation, according
2
to our definition, by noting that A = C = 1 and B = 0 imply B − 4AC = −4 < 0.
As an example of a physical situation where this equation arises, consider the tem-
perature u(x, y) in a rectangular metal plate which is insulated on the top and bottom,
so that heat cannot flow in the z−direction. If the temperatures on all four edges of the
rectangle are specified, then as t → ∞, the temperature in the interior of the rectangular
plate will approach the solution of equation (10.36).
The method of separation of variables can be successfully exploited for Laplace’s
equation in polar coordinates. One particularly important case is a circular disk. To be
specific, let us take the disk to have radius 1 and centered at the origin. Consider the
Dirichlet boundary value problem

2
2
∆u = 0, x + y < 1 (10.37)
2
2
u = h, x + y = 1, (10.38)
so that the function u(x, y) satisfies the Laplace equation on the unit disk and satisfies
the specified Dirichlet boundary conditions on the unit circle. For example, u(x, y) might
represent the displacement of a circular drum that is attached to a wire of height

h(x, y) = h(cos θ, sin θ) ≡ h(θ), 0 ≤ θ ≤ 2π,

above each point (x, y) = (cos θ, sin θ) on the unit circle.
The rectangular separable solutions are not particularly helpful in this situation. The
fact that we are dealing with a circular geometry inspires us to adopt polar coordinates

p y
2
2
x = r cos θ, y = r sin θ or r = x + y , θ = arctan ,
x
and write the solution u(r, θ) as a function thereof.

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