If $u$ is only continuous and satisfies Mean value property , is it true that $u$ is harmonic in $\Omega \subset \mathbb{R}^n$ . $\Omega$ is bounded and open. What basically here should I know to prove it . Hints are appreciated . Thanks
Mean value property satisfied of continuous functions.
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0What is $\Omega$? What exactly do you intend by mean value property? – 2012-07-11
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0@DavideGiraudo : i've edited. – 2012-07-11
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0Roughly speaking, you need to approximate $u$ by mollifiers, and then use the Mean Value Property to show that $u$ is harmonic. I learned this many years ago, and I have no reference right now. – 2012-07-11