# Generalized mean value property for the Poisson equation

It is well known that solutions to the Laplace equation in a region $$\Omega\subseteq\mathbb{R}^n$$, $$\nabla^2u=0$$ satisfy the mean value property, namely for all $$x\in\Omega$$, and for all $$r>0$$,

$$u(x) = \frac{1}{\left|B(x,r)\right|}\int\limits_{B(x,r)}u\ dV = \frac{1}{\left|\partial B(x,r)\right|}\int\limits_{\partial B(x,r)}u\ dS$$

where $$B(x,r)$$ is a ball of radius $$r$$ around $$x$$ (we assume $$B(x,r)\subseteq \Omega$$).

I was wondering whether there is a generalized property for solutions to the Poisson equation, namely suppose $$\nabla^2u=f$$ in some region $$\Omega\subseteq\mathbb{R}^n$$.

Can we find some function $$w$$ in $$\Omega$$ such that for all $$x\in\Omega$$ and for all $$r>0$$, the following is true?

$$u(x) = \frac{\int\limits_{B(x,r)}uw\ dV}{\int\limits_{B(x,r)}w\ dV} = \frac{\int\limits_{\partial B(x,r)}uw\ dS}{\int\limits_{\partial B(x,r)}w\ dS}$$

The only thing I know is that we are constrained to the fact that if $$\nabla^2u \geq 0$$, then

$$u(x) \leq \frac{1}{\left|B(x,r)\right|}\int\limits_{B(x,r)}u\ dV = \frac{1}{\left|\partial B(x,r)\right|}\int\limits_{\partial B(x,r)}u\ dS$$ and the other way around if $$\nabla^2u\leq 0$$.

Note that in your wish equality would imply that if $$u$$ is constant on a sphere, it is also constant on the whole ball. This is certainly not true.
I think what you're looking for is the Feynman-Kac formula, which says $$u(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}u \;dS+\int_{B(x,r)}f\;d\mu,$$ where the measure $$\mu$$ can be expressed by a Brownian motion $$W$$ and its exit time $$\tau$$ from the ball of radius $$r$$: $$\int_{B(x,r)}f\;d\mu=\mathbb{E}\int_0^{\tau}f(x+W_t)\;dt.$$
• Strictly speaking for answering your question random processes are not needed, you can just take my statement as the existence of a measure $\mu$ (depending on $x$ and $r$) such that this property holds. The relation to random processes is nontrivial, and very much depends on your background in stochastic analysis, so I would just suggest to look up Feynman-Kac and go from there.