Given any linear second order partial differential equation. I would like to know the steps to follow in order to obtain the mean value property of the equation. For example, I was studying a book by David Gilbarg page 27 where they have proved mean value property for the Laplace Equation \begin{align} \nabla^{2}u=0\quad \text{in}\quad \Omega \end{align} As \begin{align} u(y)=\dfrac{1}{n\omega_{n}R^{n-1}}\int_{\partial B} u\quad\mathrm{ds} \end{align}

My question is their mathematical formulation to use in order to arrive at such property. I have parabolic PDE where I want to apply the methodology.

  • $\begingroup$ Have you looked at the proof and tried to adapt it? That’s your formulation.. $\endgroup$ – DaveNine Mar 16 at 22:24
  • $\begingroup$ I have looked at the proof. The thing confusing me is that it looks like the property is only for the harmonic function. Wanted the mathematical formulation so I obtain my own mean value for my PDE to use. Thanks a lot @DaveNine $\endgroup$ – Mafeni Alpha Mar 17 at 11:16
  • $\begingroup$ Indeed, it is equation specific. You can check up on Evans on a mean value property for the heat equation, but there is no formulaic way to create one for any PDE. In fact, this is obviously not guaranteed. For hyperbolic pde information travels at a completely different way. $\endgroup$ – DaveNine Mar 17 at 22:27
  • $\begingroup$ Once again thanks @DaveNine. Wanted the mean value to use it on proving uniqueness using the method of maximum principle. I think I just have to look at other methods in order to achieve my objective $\endgroup$ – Mafeni Alpha Mar 18 at 9:31

Figured it out. The function $u(y)$ is just the average of a function $u$ over a sphere.


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