change-var-2up23

change-var-2up23 - Math 425 Spring 2011 Jerry L Kazdan PDE...

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Unformatted text preview: Math 425, Spring 2011 Jerry L. Kazdan PDE: Linear Change of Variable Let x : =( x 1 , x 2 ,..., x n ) be a point in R n and consider the second order linear partial differ- ential operator Lu : = n ∑ i , j = 1 a i j ∂ 2 u ∂ x i ∂ x j , (1) where the coefficient matrix A : = ( a i j ) is constant. Since for functions whose second derivatives are continuous we know that ∂ ∂ x i parenleftbigg ∂ ∂ x j parenrightbigg = ∂ ∂ x j parenleftbigg ∂ ∂ x i parenrightbigg we may (and will) assume that A is a symmetric matrix: A = A * . In these brief notes we obtain a useful formula for how L changes if we make the linear change of variable y = Sx where ( S : = s k ℓ ) is a constant matrix. Written in coordinates this means that y k = n ∑ ℓ = 1 s k ℓ x ℓ , where k = 1 ,..., n . F IRST G OAL : Compute L in these new y coordinates. This is straightforward (even boring) if you just keep calm and don’t make copying errors. By the chain rule ∂ u ∂ x j = n ∑ k = 1...
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