# hw9solns.pdf - APM346 HOMEWORK 9 1 Let Ω ⊂ Rn be open...

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APM346 HOMEWORK 91. Let ΩRnbe open and bounded. ANeumann eigenvalueof-Δ is a numberλRsuch that the equation-ΔX=λXin Ω∂u∂n|Ω= 0,has a nontrivial solutionX. Such a solution is called aeigenfunctionof-Δ. The setof Neumann eigenvalues of-Δ is called itsNeumann spectrum, sometimes denotedspecN(-Δ).Determine the Neumann spectrum of-Δ on the rectangle [0, Lx]×[0, Ly]. You mayassume without proof that every Neumann eigenvalue corresponds to an eigenfunctionof the formX=A(x)B(y).Solution.Substitute the ansatzX=A(x)B(y) into the equation to getA00A+B00B=-λ,so there exists constantsμxandμysuch thatA00A=-μx,B00B=-μy,λ=μx+μy.We obtainA00+μxA= 0,A0(0) =A0(Lx) = 0,B00+μyB= 0,B0(0) =B0(Ly) = 0.ThusμxandAare Neumann eigenvalues and eigenfunctions of-d2dx2on the interval[0, Lx], so there exists somenx∈ {0,1,2, . . .}such thatμx= (nxπLx)2andA(x) =Ccos(nxπxLx). Similarly forB, with “x” replaced by “y”. Thus the separated solutions

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