Unformatted text preview: ion of Wλ is called the multiplicity of
the eigenvalue λ. The eigenvalue problem consist of ﬁnding the eigenvalues λ,
and a basis for each nonzero eigenspace.
Once we have solved the eigenvalue problem for a given region D in the
(x, y )plane, it is easy to solve the initial value problem for the heat equation
or the wave equation on this region. To do so requires only that we substitute
the values of λ into the equations for g . In the case of the heat equation,
2 g (t) = c2 λg (t) ⇒ g (t) = (constant)ec λt , while in the case of the wave equation,
√
g (t) = c2 λg (t) ⇒ g (t) = (constant) sin(c −λ(t − t0 )).
In the second case, the eigenvalues determine the frequencies of a vibrating
drum which has the shape of D.
Theorem. All of the eigenvalues of ∆ are negative, and each eigenvalue has
ﬁnite multiplicity. The eigenvalues can be arranged in a sequence
0 > λ1 > λ2 > · · · > λ n > · · · ,
142 with λn → −∞. Every wellbehaved function can be represented as a convergent
sum of eigenfunctions.2
Although the theorem is reassuring, it is...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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