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Prove the 2nd form of the mean value property: if ueC?(2) is harmonic, then u(x)= m(B(x,r)) u(y)dy Bx.r) for each ball B(x, r)EN, where m(B(x,r))=[mcdy denotes the volume of the ball B(x,r). Hint: use the coarea formula f(x)dS for any continuous and integrable function f.

Prove the 2nd form of the mean value property: if ueC?(2) is harmonic, then u(x)= m(B(x,r)) u(y)dy Bx.r) for each ball B(x, r)EN, where m(B(x,r))=[mcdy denotes the volume of the ball B(x,r). Hint: use the coarea formula f(x)dS for any continuous and integrable function f.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Prove the 2nd form of the mean value property: if ue Cc*2) is harmonic, then u(x)= m(B(x,r)) u(y) dy (x.r) for each ball B(x, r)EN, where m(B(x,r))=/mc dy denotes the volume of the ball B(x, r). Hint: use the coarea formula B(x.r f(x)dx%3D f(x)dS )dr for any continuous and integrable function f.
Transcribed Image Text:4. Prove the 2nd form of the mean value property: if ue Cc*2) is harmonic, then u(x)= m(B(x,r)) u(y) dy (x.r) for each ball B(x, r)EN, where m(B(x,r))=/mc dy denotes the volume of the ball B(x, r). Hint: use the coarea formula B(x.r f(x)dx%3D f(x)dS )dr for any continuous and integrable function f.
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