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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 set $B(x,r)$ s.t.

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

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Similar equations are studied in the field of Fredhold equations and Volterra equations, and more broadly speaking in the study of integral equations. They often, though not always, arise in the context of a variational formulation of a given PDE. The particular equation that you have written can be thought of (for suitable kernels $K$) as a fixed-point problem for a Fredholm integral operator.

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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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