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Chapter 10

Laplace’s Equation

10.1 Solved Problems

Exercise 198. Is the solution of the following Dirichlet problem unique?

u xx + u yy = q(x, y), −∞ < x < ∞, y > 0
u(x, 0) = f(x)

Solution: Let u and v both be solutions of the Dirichlet problem. Let w be the
difference of these solutions. w satisfies the problem

w xx + w yy = 0, −∞ < x < ∞, y > 0
w(x, 0) = 0.

Since w = cy is a solution. We conclude that the solution of the Dirichlet problem
is not unique.

Exercise 199. Is the solution of the following Dirichlet problem unique?

u xx + u yy = q(x, y), −∞ < x < ∞, y > 0
2
2
u(x, 0) = f(x), u bounded as x + y → ∞
Solution: Let u and v both be solutions of the Dirichlet problem. Let w be the
difference of these solutions. w satisfies the problem

w xx + w yy = 0, −∞ < x < ∞, y > 0
2
2
w(x, 0) = 0, w bounded as x + y → ∞.
We solve this problem with a Fourier transform in x.

2
−ω ˆw + ˆw yy = 0, ˆ w(ω, 0) = 0, ˆ w bounded as y → ∞
(
c cosh ωy + c sinh(ωy), ω 6= 0
2
1
ˆ w =
c + c y, ω = 0
2
1
ˆ w = 0
w = 0

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