Fix a positive integer n. The Laplacian p of a twice differentiablefunction p on Rn is

Chapter 6, Problem 30

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Fix a positive integer n. The Laplacian p of a twice differentiablefunction p on Rn is the function on Rn defined byp D @2p@x21CC@2p@x2n:The function p is called harmonic if p D 0.A polynomial on Rn is a linear combination of functions of theform x1m1 xnmn , where m1;:::;mn are nonnegative integers.Suppose q is a polynomial on Rn. Prove that there exists a harmonicpolynomial p on Rn such that p.x/ D q.x/ for every x 2 Rn withkxk D 1.[The only fact about harmonic functions that you need for this exerciseis that if p is a harmonic function on Rn and p.x/ D 0 for all x 2 Rnwith kxk D 1, then p D 0.]Hint: A reasonable guess is that the desired harmonic polynomial p is ofthe form q C .1 kxk2/r for some polynomial r. Prove that there is apolynomial r on Rn such that q C .1 kxk2/r is harmonic by definingan operator T on a suitable vector space byT r D .1 kxk2/rand then showing that T is injective and hence surjective.