Harmonic Functions

**Jose27** · October 12th 2009, 09:35 AM

Let $f: \Omega \subset \mathbb{R} ^n \longrightarrow \mathbb{R}$ such that $f \in L^1 ( \Omega )$ and for all $x \in \Omega$ and $R_x$ such that $\overline{B_{R_x} (x)} \subset \Omega$ we have $f(x)=\frac{1}{ \vert B_{R_x}(x) \vert } \int_{B_{R_x}(x)} \ f$ . Is $f$ harmonic in $\Omega$ ?

I know the result is true if $f \in C^2( \Omega )$ , but I don't know how to argue that if $f \in L^1 ( \Omega )$ then it's twice differentiable.

**Jose27** · April 24th 2010, 08:57 PM

Originally Posted by Jose27

Let $f: \Omega \subset \mathbb{R} ^n \longrightarrow \mathbb{R}$ such that $f \in L^1 ( \Omega )$ and for all $x \in \Omega$ and $R_x$ such that $\overline{B_{R_x} (x)} \subset \Omega$ we have $f(x)=\frac{1}{ \vert B_{R_x}(x) \vert } \int_{B_{R_x}(x)} \ f$ . Is $f$ harmonic in $\Omega$ ?

I know the result is true if $f \in C^2( \Omega )$ , but I don't know how to argue that if $f \in L^1 ( \Omega )$ then it's twice differentiable.

Okay, hadn't given this one much thought. Here's one proof for continous functions under slightly different assumptions; and although it is an improvement from the $C^2$ case I'm still interested in the general case.

Let $\Omega \subseteq \mathbb{R} ^n$ open and $f: \Omega \rightarrow \mathbb{R}$ be such that for all $x\in \Omega$ and $r>0$ such that $\overline{B} _r(x) \subset \Omega$ we have $f(x)= \frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)} f(y)dy$ .

Let $\eta : \mathbb{R} ^n \rightarrow \mathbb{R}$ be given by $\eta (x)= ce^{\frac{1}{\| x \| ^2-1}}$ where $c>0$ is such that $\int_{\mathbb{R} ^n} \eta =1$ . Define $\eta_{ \varepsilon } (x)= \varepsilon ^{-n} \eta \left( \frac{x}{\varepsilon } \right)$ . It is a standard result that if $f\in L_{loc}^1 (\Omega )$ then $f_{\varepsilon }:= f\ast \eta _{\varepsilon } \in C^{\infty} ( \Omega _{\varepsilon } )$ where $\Omega_{ \varepsilon } := \{ x\in \Omega : d(x,\partial \Omega )> \varepsilon \}$ .

With this in mind, let $f: \Omega \rightarrow \mathbb{R}$ be continous then:

$f_{\varepsilon } (x)= \int_{B_{\varepsilon }(0)} f(x-y) \eta_{\varepsilon }(y)dy=\int_{0}^{\varepsilon } \eta _{\varepsilon }(r) \int_{ \partial B_{\varepsilon }(x)} f(y)dSydr$ $= \int_{0}^{\varepsilon } \eta _{\varepsilon } (r) f(x) |\partial B_{\varepsilon }(x)|dr=f(x) \int_{B_{\varepsilon }(x) }\eta _{\varepsilon } (y)dy=f(x)$

This proves that $f$ is harmonic in $\Omega$ . Now the issue here, if we wanted to generalize this procedure to $f$ only integrable (or locally so), is that $\partial B_r(x)$ has measure zero so our change into "polar coordinates" can not be applied as we did here.

Any comments or suggestions are appreciated.

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