# Moreover using the integral formulae 2 cos n cos md

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Unformatted text preview: ab = 1, a, b > 0. Hint: One can eliminate b by setting a = 1/b and then ﬁnd the minimum of the function g (a) = π a 2 2 + (πa) , when a > 0. 5.7.2. Let D be a ﬁnite region in the (x, y )-plane bounded by a smooth curve ∂D. Suppose that the eigenvalues for the Laplace operator ∆ with Dirichlet boundary conditions on D are λ1 , λ2 , . . . , where 0 > λ1 > λ2 > · · · , 2 Note the similarity between the statement of this theorem and the statement of the theorem presented in Section 4.7. In fact, the techniques used to prove the two theorems are also quite similar. 3 A few more cases are presented in advanced texts, such as Courant and Hilbert, Methods of mathematical physics I , New York, Interscience, 1953. See Chapter V, §16.3. 143 each eigenvalue having multiplicity one. Suppose that φn (x, y ) is a nonzero eigenfunction for eigenvalue λn . Show that the general solution to the heat equation ∂u = ∆u ∂t with Dirichlet boundary conditions (u vanishes on ∂D) is ∞ bn φn (x, y )eλn t , u(x, t) = n=1 where the bn ’s are arbitrary constants. 5.7.3. Let D be a ﬁnite region in the (x, y )-...
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## This document was uploaded on 01/12/2014.

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