Comment # 18 on bug 973599 from Dr. Werner Fink

To get DISPLAY and XAUTHORITY this requires some tricky bash environment which
depends on the display manager as well as on the sudo configuration.

In my /etc/sudoers.d/werner I'm using

  env_keep+="LS_COLORS LS_OPTIONS DISPLAY"

then in /root/.bashrc the lines

 if test -n "$SUDO_USER" ; then
     SUDO_HOME=$(eval echo ~$SUDO_USER)
     if test -n "$SUDO_HOME" -a -s "$SUDO_HOME/.bashrc.root"; then
         export BASH_ENV="$SUDO_HOME/.bashrc.root"
         . "$SUDO_HOME/.bashrc.root"
     fi
 fi

and in my $HOME/.bashrc.root

 if test "${DISPLAY%[0-9]*}" = : ; then
     for auth in /var/lib/xdm/authdir/authfiles/A${DISPLAY}-* /run/sddm/\{* ;
do
         test -e "$auth" || continue
         XAUTHORITY=$auth
         break
     done
     if test -e "$XAUTHORITY" ; then
         export XAUTHORITY
     else
         unset XAUTHORITY
     fi
 fi

... and if gdm is used this may adopted to the locations where gdm stores its
authority files