MittelwertEigenschaft

The MittelwertEigenschaft, often translated as the mean value property, is a concept primarily found in the context of partial differential equations and harmonic functions. It states that for a harmonic function, its value at any point within a domain is equal to the average of its values over any sphere or ball centered at that point, provided the sphere or ball is entirely contained within the domain. Mathematically, if $u$ is a harmonic function and $B(x, r)$ is a ball of radius $r$ centered at $x$, then $u(x) = \frac{1}{\text{Vol}(B(x, r))} \int_{\partial B(x, r)} u(s) \, dS$, where $\partial B(x, r)$ denotes the boundary of the ball and $dS$ is the surface element. Alternatively, considering a ball $B(x, r)$, the property states that $u(x) = \frac{1}{\text{Vol}(B(x, r))} \int_{B(x, r)} u(y) \, dV$, where $dV$ is the volume element. This property is a fundamental characteristic of harmonic functions and serves as a key tool in their analysis. It implies that harmonic functions cannot attain their maximum or minimum values in the interior of their domain; such extrema must occur on the boundary. The MittelwertEigenschaft is a defining characteristic that distinguishes harmonic functions from other types of functions.