Let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^2\) of class \(C^{1,\eta}\) for some \(\eta\in]0,1[\). Let \(\lambda\in]0,+\infty[\), \(\epsilon\in]0,+\infty[\). Let \[ {\mathcal{L}}_\epsilon\equiv-{\operatorname{div}}\left[A(x/\epsilon)\nabla+V(x/\epsilon)\right] +B(x/\epsilon)\nabla+c(x/\epsilon)+\lambda I \] be a uniformly elliptic differential operator of order \(2\), with periodic coefficients \(A\), \(V\), \(B\), \(c\) with periodicity cell equal to \([0,1[^2\). Under suitable assumptions, the authors prove \(\epsilon\)-independent Sobolev and Hölder a priori estimates for the weak solution \(u_\epsilon\) of the Dirichlet problem \[ {\mathcal{L}}_\epsilon(u_\epsilon)={\operatorname{div}} (f)+F\quad\text{in}\ \Omega\,, \quad u_\epsilon=g\quad\text{on}\ \partial\Omega\,, \] for some data \(f\) and \(F\) in a Lebesgue space in \(\Omega\) and for some \(g\) that belongs to a Besov or to a Hölder/Schauder space on \(\partial\Omega\), respectively. The authors also prove an estimate on the Green function and an estimate of the rate of convergence of \(u_\epsilon\) to a limiting solution as \(\epsilon\) tends to \(0\). The paper under review extends to the \(2\)-dimensional case previous results of Q. Xu [J. Math. Anal. Appl. 438, No. 2, 1066–1107 (2016; Zbl 1337.35009)], who had considered the \(d\)-dimensional case with \(d\geq 3\).