Let f be C2 on R3 and satisfy Laplace's equation V2f = 0. Such functions are called harmonic. %3D (a) Applying Green's formulas to f and 1 g(x, y, z) x2 + y2 + z2 over R = {(x, y, z) | ɛ < ||(x, y, z)||

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Chapter2: Second-order Linear Odes
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1. Let f be C2 on R3 and satisfy Laplace's equation V2f = 0. Such functions are called harmonic.
(a)
Applying Green's formulas to f and
1
g(x, y, z) =
x2 + y2 + z2
over R = {(x, y, z) | e < ||(x, Y, z)|| < r}, show that the mean values of f on the spheres
||(x, y, z)||
= r and ||(x, y, z)|| =
€ are equal.
(b)
the value of f at the origin. (Side remark: there is nothing special about the origin here.
Applying the result to f(x, y, z) = f((x+ a, y + b, z + c)), we see that the mean value of a
harmonic function over any sphere is its value at the center of the sphere.)
Conclude that the mean value of f on any sphere centered at the origin is equal to
Transcribed Image Text:1. Let f be C2 on R3 and satisfy Laplace's equation V2f = 0. Such functions are called harmonic. (a) Applying Green's formulas to f and 1 g(x, y, z) = x2 + y2 + z2 over R = {(x, y, z) | e < ||(x, Y, z)|| < r}, show that the mean values of f on the spheres ||(x, y, z)|| = r and ||(x, y, z)|| = € are equal. (b) the value of f at the origin. (Side remark: there is nothing special about the origin here. Applying the result to f(x, y, z) = f((x+ a, y + b, z + c)), we see that the mean value of a harmonic function over any sphere is its value at the center of the sphere.) Conclude that the mean value of f on any sphere centered at the origin is equal to
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