
> 
> **Question.** Is there a more general set of equations that satisfy mean value properties, similar to the Laplacian and heat equation? 
> 
> 
> 


For example, finding some kernel $K(x,y)$ and a set $B(x,r)$ such that


$$u(x)=\int\_{B(x,r)} u(y)K(x,y)dy.$$


Even existence of such a $K$ and a $B$ is good enough for me; eg. showing that the Fredhold integral equation has a fixed point that also solves a more general pde. Or even more generally the existence of some measure $\mu$ that gives:


$$u(x)=\int\_{B(x,r)} u(y)\mu(dy).$$


Can this be approached as a fixed point problem? 


For example, for $Lu=(\Delta +c)(u)$ there is an interesting MVP


$$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4\pi \rho^2}\int\_{\partial B(\xi,\rho)} u(x)d\sigma(x).$$



reference: modified mean value property for more general pdes
It generalizes for many elliptic and parabolic pdes:


1)"Spherical Means for Pdes"
edited by K. Karl Karlovich Sabelfeld, I. A. Shalimopva


2)"Spherical and plane integral operators for PDEs : construction, analysis, and applications", Karl K. Sabelfeld, Irina A. Shalimova, 


