Consider the Green's problem $$\Delta G(x,y|s,t)=\delta(x-s)\delta(y-t)\ \ on\ \ R\\ G=0 \ \ on \ \ \partial R $$ Check if $G(x,y|s,t)=G(s,t|x,y)?$
Can not we say that since the differential operator is self adjoint, $G$ is symmetric? Or it depends on the location of $s,t$ if it is in $R$, on $\partial R$, or outside $R \cup \partial R.$
Thanks in advance for any help.