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Math Help - Harmonic Functions

  1. #1
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    Harmonic Functions

    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.
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  2. #2
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    Quote Originally Posted by Jose27 View Post
    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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