Mean property

Mean value property in mathematics is the property of a function that the function value and the averaged function value in a sphere around this point correspond to each other.

A function that fulfills the mean value property is automatically harmonic and smooth, i.e. in . ${\displaystyle C^{\infty }(\Omega )}$

definition

Be . A function fulfills the mean value property if and only if for all and all that suffice ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon \Omega \to \mathbb {R} }$ ${\displaystyle x\in \Omega }$ ${\displaystyle r>0}$ ${\displaystyle B_{r}(x)\subset \Omega }$

${\displaystyle f(x)={\frac {1}{\|B_{r}\|}}\int _{B_{r}(x)}f(y)dy={\frac {1}{\|\partial B_{r}\|}}\int _{\partial B_{r}(x)}f(y)dS(y)}$

applies. And stand for the volume or the surface of the sphere with radius . ${\displaystyle \|B_{r}\|}$ ${\displaystyle \|\partial B_{r}\|}$ ${\displaystyle r}$

The integrals with a prefactor are averaged integrals , which are often also noted as a crossed out integral.

Sufficient condition

Both requirements, i.e. the equality of the function value with the averaging over the entire sphere or that over its surface, are equivalent. This follows from the formulas for surface and volume of the -dimensional sphere, because if the mean value property with the surface integral is true, is ${\displaystyle n}$

${\displaystyle {\frac {1}{\|B_{r}\|}}\int _{B_{r}(x)}f(y)dy={\frac {1}{\|B_{r}\|}}\int _{0}^{r}\underbrace {\int _{\partial B_{s}(x)}f(y)dy} _{=f(x)\|\partial B_{s}\|}ds={\frac {1}{\|B_{r}\|}}f(x)\|B_{r}\|=f(x)}$

The opposite is true if the mean value property applies to the integral over the full sphere, according to the main theorem

${\displaystyle {\frac {1}{\|\partial B_{r}\|}}\int _{\partial B_{r}(x)}f(y)dy={\frac {1}{\|\partial B_{r}\|}}({\frac {d}{ds}}\underbrace {\int \int _{\partial B_{s}}f(y)dyds} _{=f(x)\|B_{s}\|}){\Big |}_{s=r}={\frac {1}{\|\partial B_{r}\|}}f(x)\|\partial B_{r}\|=f(x)}$

So it is enough to prove one of the conditions.

Attenuated mean value property

When studying sub- and superharmonic functions , a weakened formulation of the mean value property is used, in which the equals sign is replaced by less or greater than:

${\displaystyle \forall x\in \Omega :f(x)\leq {\frac {1}{\|B_{r}\|}}\int _{B_{r}(x)}f(y)dy}$

Discrete mean property

In the numerics of partial differential equations one speaks of the discrete mean value property in connection with the discretization of the Laplace operator : By forming the second centered difference quotient one arrives at the approximation

${\displaystyle \Delta _{h}f(x)\approx {\frac {1}{h^{2}}}\left(f(x-h)-2f(x)+f(x+h)\right)}$

The fact that an average property also applies here can be seen directly by inserting a discrete harmonic function for which the following applies: ${\displaystyle \Delta _{h}f=0}$

${\displaystyle 0=f(x-h)-2f(x)+f(x+h)\Leftrightarrow f(x)={\frac {1}{2}}(f(x-h)+f(x+h))}$

literature

• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2002, ISBN 0-8218-0772-2 ( Graduate studies in mathematics 19).