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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

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

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  • $\begingroup$ Crossposted on MathSE: math.stackexchange.com/questions/2069041/…. $\endgroup$ – Alex M. Dec 23 '16 at 20:38
  • $\begingroup$ It turned out to be a more difficult problem than I thought. So I switched it here. I cannot delete the MathSE now because it has an answer. $\endgroup$ – OOESCoupling Dec 25 '16 at 19:04
  • $\begingroup$ No harm done, but in general please keep in mind that crossposting is not well received in the StackExchange communities (it is interpreted as lack of patience). Furthermore, it is recommended that you wait for several days, or even a week, before deciding to crosspost. $\endgroup$ – Alex M. Dec 25 '16 at 21:35
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$p$-Laplace equations and the $\infty$-Laplace equation have a sort of mean value property, but it presumably is not of the kind that you seek.

For $\infty$-Laplacian, $2u(x) = \max u + \min u$ with the maximum and the minimum taken over a ball with center at $x$, and with a small error term that goes to zero (at a particular rate, see reference below) as the radius of the ball goes to zero.

For $p$-Laplace equations you have a linear combination of the mean value property for the usual 2-Laplacian and of the mean value property for the $\infty$-Laplacian.

See for example http://users.jyu.fi/~miparvia/Julkaisuja/final_MeanValues.pdf

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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,

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