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In his answer to an MO question: "Why is the Laplacian ubiquitous?", Terry Tao says that

The Laplacian of a function $u$ at a point $x$ measures the average extent to which the value of $u$ at $x$ deviates from the value of $u$ at nearby points to $x$ (cf. the mean value theorem for harmonic functions).

Would anybody elaborate (probably with examples) how the mean value theorem for harmonic functions relates to the point he makes about the Laplacian?


marked as duplicate by Willie Wong, Claude Leibovici, Adrian, Lee Mosher, hardmath Nov 21 '16 at 15:25

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    $\begingroup$ See here: math.stackexchange.com/questions/50274/… $\endgroup$ – Christian Blatter Nov 18 '16 at 15:18
  • $\begingroup$ The mean value theorem for harmonic functions says that harmonic functions don't deviate from their local average, which matches up with their Laplacian being zero. When it is nonzero, there is deviation between them, and the sign tells you which one is bigger. $\endgroup$ – Ian Nov 18 '16 at 15:29

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