Let $U \subset \mathbb{R}^n$ be an open set and suppose that $u$ is harmonic on $U$. The theorem 2 of Evan's Partial Differential Equations 2nd edition makes a connection between the pointwise and average behaviour of harmonic functions on $U$. In the theorem itself the closure of the ball in question need not be inside of $U$. Namely,
(Mean-value formulas for the Laplace's equation) If $u \in C^2(U)$ is harmonic, then $$u(x) = \frac{\int_{\partial B(x, r)}udS}{|\partial B(x, r)|} = \frac{\int_{B(x, r)}u dy}{| B(x, r)|}$$ for each ball $B(x, r) \subset U$.
What I am wondering is that if $x \in U$ and $r > 0$ such that $B(x, r)\subset U$, do we then also need to have that $\overline{B(x, r)}\subset U$ for us to be able so use the said formula? I am wondering this since do we not have some general PDE solutions which are harmonic in a punctured space/set, so that you could take a suitable open set whose boundary crosses such a punctured point? Or could we have a harmonic function which is positive on $U$ but vanishes on $\partial U$? Consequently, if we are not interested in the surface integral but only on the averages over the balls, is it then okay if $\overline{B(x, r)}\not\subset U$, or more concisely $\partial B(x, r)\not\subset U$?