# Such a theory can be developed for an arbitrary

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Unformatted text preview: for (x, y ) ∈ ∂D. The same method can be used to treat the wave equation ∂2u = c2 ∆u, ∂t2 141 (5.27) with the Dirichlet boundary condition, u(x, y, t) = 0 for (x, y ) ∈ ∂D. This time, substitution of u(x, y, t) = f (x, y )g (t) into (5.27) yields f (x, y )g (t) = c2 (∆f )(x, y )g (t), or 1 1 g (t) = (∆f )(x, y ). c2 g (t) f (x, y ) Once again, both sides must equal a constant λ, and we obtain 1 1 g (t) = λ = (∆f )(x, y ). c2 g (t) f (x, y ) This separates into g (t) = λg (t) and (∆f )(x, y ) = λf (x, y ), in which f is once again subject to the boundary condition, f (x, y ) = 0 for (x, y ) ∈ ∂D. In both cases, we must solve the “eigenvalue problem” for the Laplace operator with Dirichlet boundary conditions. If λ is a real number, let Wλ = {smooth functions f (x, y ) : ∆f = λf, f |∂D = 0}. We say that λ is an eigenvalue of the Laplace operator ∆ on D if Wλ = 0. Nonzero elements of Wλ are called eigenfunctions and Wλ itself is called the eigenspace for eigenvalue λ. The dimens...
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