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642 CHAPTER 18 The Potential Equation

2
∇ u = 0on D
and
u(x, y) = f (x, y) for (x, y) in ∂ D

with f (x, y) as a given function. The function f is called boundary data for D.

SECTION 18.1 PROBLEMS

4
2
2
1. Show that if f and g are harmonic on D so are f + g (c) x − 6x y + y 4
3
and, for any constant c, cf . (d) 4x y − 4xy 3
−y
y
2. Show that the following functions are harmonic (on the (e) sin(x)(e + e )
y
−y
entire plane, if D is not specified). (f) cos(x)(e − e )
(g) e −x cos(y)
2
2
3
(a) x − 3xy 2 (h) ln(x + y ) if D is the plane with the origin
2
(b) 3x y − y 3 removed.
18.2 Dirichlet Problem for a Rectangle
The region D exerts a great influence on our ability to explicitly solve a Dirichlet problem, or
even whether a solution exists. Some regions admit solutions by Fourier methods. In this and the
next section, we will treat two such cases: that D is a rectangle or disk in the plane.
Let D be the solid rectangle consisting of points (x, y) with 0 ≤ x ≤ L,0 ≤ y ≤ K. We will
solve the Dirichlet problem for D.
This problem can be solved by separation of variables if the boundary data is nonzero on
only one side of D. We will illustrate this for the case that this is the upper horizontal side of D.
The problem in this case is
2
∇ u = 0on D,
u(x,0) = 0for 0 ≤ x ≤ L,
u(0, y) = 0for 0 ≤ y ≤ K,
u(L, y) = 0for 0 ≤ y ≤ K,
and
u(x, K) = f (x) for 0 ≤ x ≤ L.

Figure 18.1 shows D and the boundary data. Let u(x, y) = X(x)Y(y) in Laplace’s equation
to obtain
X Y
=− =−λ
X Y
or
X + λX = 0 and Y − λY = 0.

From the boundary conditions, X(0) = X(L) = Y(0) = 0, so the problems for X and Y are

X + λX = 0; X(0) = X(L) = 0

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October 14, 2010 15:27 THM/NEIL Page-642 27410_18_ch18_p641-666