Mean Value property (harmonic functions) with a source?

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Mean Value property (harmonic functions) with a source??

Homework Statement



I understand that the heat equation may yield ∂u/∂t=0 on the LHS and on the RHS we may still have Uxx+Q where Uxx is partial with respect to x twice and Q is a heat source. U in our case may be the temperature function.

Now my teacher introduces the mean value property for harmonic functions and I ask, when there is a heat source the mean value property breaks down, is this because our RHS is not a harmonic function or am I wrong in assuming the property doesn't still hold. My reasoning is that if we take a cross section of let's say a cylinder with a wire (sourse) going through the middle and we take a point close to the source, create a ball around it, then due to the conservation of energy we will find that the point has a temperature hotter than the average temperature along the boundary of the ball we created.

Much input is needed/appreciated as my teacher wasn't able to address my question.

Homework Equations



1.4 Mean-Value Property: If u is harmonic on B(a,r), then u equals the average of u over ∂B(a,r).

The Attempt at a Solution



Hours of thinking
 

Answers and Replies

  • #2
Dick
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Nothing wrong with your thinking. If a function is harmonic then Uxx=0. If your equation is Uxx+Q=0 and Q is nonzero then you have a heat source and U is no longer harmonic. So the mean value property no longer applies. U is harmonic only where Q=0.
 

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