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Unformatted text preview: gives a decomposition into eigenfunctions which solve the
Such a theory can be developed for an arbitrary bounded region D in the
(x, y )-plane bounded by a smooth closed curve ∂D. Let V denote the space of
smooth functions f : D → R whose restrictions to ∂D vanish. It is important
to observe that V is a “vector space”: the sum of two elements of V again lies
in V and the product of an element of V with a constant again lies in V .
We deﬁne an inner product , on V by
f, g = f (x, y )g (x, y )dxdy.
D Lemma. With respect to this inner product, eigenfunctions corresponding to
distinct eigenvalues are perpendicular; if f and g are smooth functions vanishing
on ∂D such that
∆f = λf, ∆g = µg, (5.32) then either λ = µ or f , g = 0.
The proof of this lemma is a nice application of Green’s theorem. Indeed, it
follows from Green’s theorem that
dx + f
D D ∂
[f (∂g/∂x)] −
[−f (∂g/∂y )] dxdy
∂y ∂f ∂g
∂ f ∂g
∂x ∂x ∂y ∂y f
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This document was uploaded on 01/12/2014.
- Winter '14