Let $D:= B(0,R)$\ $B(0,1)$ for $R>1$ such that $\Delta H=0$ in $D$, $H=0$ on $\partial B(0,1)$, and $H=1$ on $B(0,R)$.
I would like to find an explicit solution to this problem, however I am unsure how to go about it. Since the laplacian is zero, my first thought was that I could use the harmonic function for the two-dimensional Brownian motion, that is, $H_2(x)=\log(\|x\|)$, but I don't see how this works here.
Any help would be much appreciated.