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$u$ satisfies "mean value property locally " on $\Omega$ if for every $x\in \Omega \exists \delta=\delta (x)>0 $ such that

$u(x) \le \frac {1}{|\mu(B(x,r)|}\int_{\partial B(x,r)} u(y) dS_y$

for all $r\le \delta(x)$

Does this imply that if $u\in C^2(\Omega)$ and satisfies mean value property locally in $\Omega$ then $u$ is subharmonic ?? Any hints will be nice.

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    Yes, I am sure this is covered in standard texts on potential theory, such as Helms ("Potential Theory"). In most of potential theory, we don't require $C^2$, just upper or lower semi-continuity.2012-07-17
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    @Old John : What is the basic argument to extend local property to the global property ?2012-07-17
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    Hint: use the 2nd order Taylor expansion of u at x. The first derivatives cancel out in integration, the second derivatives contribute the Laplacian, and nothing else matters.2012-07-17
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    From memory, I think it is a connectedness argument, but I think you might have to adjust the statement of the problem above, as I think it would fail if $\Omega$ were not connected.2012-07-17
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    @LeonidKovalev : I will try and will try to get back with an answer .2012-07-17
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    Interestingly, Ransford (Potential Theory in the Complex Plane) and Hayman and Kennedy (Subharmonic functions) use the local submean property to define subharmonicity.2012-07-17
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    @LeonidKovalev : Sir , i need some more hints . I am getting something like $\epsilon \le \int_{\partial B(x,r)} Du(x) (y-x) +\int_{\partial B(x,r)}D^2 u(x) (y-x)^2 $. I doubt if i am on the right track.2012-07-18
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    @Theorem You should be aware that $Du$ is a vector, and so is $y-x$, while $D^2u$ is a matrix. Right now you are treating them as if they were numbers. It may help to spell out the vectors and matrices in coordinates $Du=(u_{x_1},\dots,u_{x_n})$, etc.2012-07-18

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