Problem A (Bonus): Prove the Mean Value Property of harmonic functions as stated in the identity in the middle of page 6

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Problem A Bonus Prove The Mean Value Property Of Harmonic Functions As Stated In The Identity In The Middle Of Page 6 1 (144.42 KiB) Viewed 40 times

Problem A (Bonus): Prove the Mean Value Property of harmonic functions as stated in the identity in the middle of page 67 of the textbook, as well as our own lecture notes, both in 2D for circles, and in 3D for spheres, with any arbitrary radius r>0 : 2D: u(x0​,y0​)=2πr1​∫Cr​(x0​,y0​)​u(x,y)ds 3D: u(x0​,y0​,z0​)=4πr21​∬Sr​(x0​,y0​,z0​)​u(x,y,z)dA Hint: A powerful strategy is to show the right hand side, as a function of the radius r, is constant. But first, use the linear substitution x=x0​+ry to convert the right hand side to an integral over the unit circle (or sphere) centered at the given point x0​, whose value at r=0 can be found very easily. Then differentiate the new integral with respect to r using chain rule, gradient of u, and the Divergence Theorem to prove it is equal to 0 . Notice that Divergence Theorem also works for closed 2D curves; the surface integral is replaced by the line integral, etc.