# Mean value property satisfied of continuous functions.

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

• What is $\Omega$? What exactly do you intend by mean value property? Jul 11 '12 at 12:24
• @DavideGiraudo : i've edited. Jul 11 '12 at 12:28
• Roughly 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. Jul 11 '12 at 12:31

One approach is to convolve with a radially symmetric mollifier and show that the convolution actually agrees with $u$ (it's better than an approximation). Intuitively, radial symmetry means that integrating over spheres gives you $u(x)$, and the for the radial integration use that the mollifier has weight one. This shows that $u$ is in fact smooth.
To show it is harmonic, an interesting approach is to use second-order Taylor approximation to show that for $C^2$ functions $u$, we have $$\Delta u(x) = \lim_{r \rightarrow 0} \frac{2}{r^2}\left(\frac{1}{|\partial B_r|}\int_{\partial B_r(x)} u - u(x)\right).$$ Applying the mean value property gives harmonicity.