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