Let $B_r(x)$ be an open ball in $\mathbb R^n$ centered at $x$ with radius $r$. In the literature, it is known that if $u \in C^2(B_{2}(0))$ and $u$ is harmonic in the sense that $\Delta^2u = 0$ everywhere, then $u$ has the mean value property, namely $$ u(0) = \frac 1{|\partial B_1(0)|}\int_{\partial B_1(0)} u(y) d\sigma_y. $$ In the above formula, it seems that the information of $u$ strictly inside the ball $B_1(0)$ does not appear in the integral.
My question is the following: Can we weaken the regulariry of $u$ inside $B_1(0)$? Let suppose that instead of $u \in C^2(B_{2}(0))$ we only assume that $$ u \in C^2(B_{2}(0) \setminus \{ 0 \}) \cap C^1(B_{1}(0)),$$ or even weaker $$ u \in C^2(B_{2}(0) \setminus \{ 0 \}) \cap C(B_{1}(0)),$$ should the above mean value property remain valid? By means of this we still assume that $\Delta^2u = 0$ everywhere except at the orinin.
Thank you.