I'm reading Jost's Partial Differential Equation Chapter 2. To point out where I'm stuck, here is the excerpt of the textbook dealing with Weyl's lemma.
(The mollifier is defined as $\rho (t) = C exp(\dfrac{1}{t^2-1})$ for $0\le t < 1$, where $C$ is a constant that makes integral of $\rho = 1$ )
So the red line is where I got stuck. To explain in detail, isn't there a ball $B$ with center in $\Omega_r$ and radius bigger than $r$ (smaller than $R/2$) such that $B\not\subset \Omega$? Even if we confine our ball $B$ to be a subset of $\Omega$, unless $B\subset \Omega_r$ holds, I have no idea why the statement holds. (Of course if $B\subset \Omega_r$, then $u_r$ is harmonic in there, so the mean value formula holds. In the case $B\subset \Omega$, $u_r$ is not guaranteed harmonic so I've tried direct integration just from the definition but it never worked.)
In the latter part of the proof, author uses family of functions $\{ u_r\}$ and deals uniform boundness and equicontinuity, therefore I suspect the mean value formula should hold for $B\subset \Omega$ in order to make the domain consistent. (Although I don't know how Mean value formula directly implies the uniform boundness. Author explains this by "fix $R$ and let $r\rightarrow 0$", but these $R$ and $r$ caused me confusion from the beginning...)
So I will be extremely grateful if somebody points out the method to make the red-lined statement work. While googling Weyl's lemma, I've seen some references regarding of functional analysis and convolutions, but I don't have any knowledge about those at this moment.
Thank you in advance! Have a nice day :)