  Since Reed conjectured in 1996 that the domination number of a connected
cubic graph of order $n$ is at most $\lceil \frac13 n \rceil$, the domination
number of cubic graphs has been extensively studied. It is now known that the
conjecture is false in general, but Henning and Dorbec showed that it holds for
graphs with girth at least $9$. Zhu and Wu stated an analogous conjecture for
2-connected cubic planar graphs.
  In this paper, we present a new upper bound for the domination number of
subcubic planar graphs: if $G$ is a subcubic planar graph with girth at least
8, then $\gamma(G) < n_0 + \frac{3}{4} n_1 + \frac{11}{20} n_2 + \frac{7}{20}
n_3$, where $n_i$ denotes the number of vertices in $G$ of degree $i$, for $i
\in \{0,1,2,3\}$. We also prove that if $G$ is a subcubic planar graph with
girth at least 9, then $\gamma(G) < n_0 + \frac{13}{17} n_1 + \frac{9}{17} n_2
+ \frac{6}{17} n_3$.
