Given a bounded and smooth domain \(\Omega\subset\mathbb{R}^N,\) the authors deal with nonnegative selfadjointness in \(L^2(\Omega)\) of the maximal realization \(T_2\) of an \(N\)-dimensional, second-order divergence form elliptic operator with diffusion coefficients that vanish on \(\partial\Omega\). By means of the singular perturbation arguments developed by the second author in [J. Math. Soc. Japan 32, 19–44 (1980; Zbl 0414.47023)], it is proved that the resolvent of \(T_2\) is given as the uniform limit \(\lim_{n\to\infty}\big(\xi+n^{-1}(-\Delta)+T_2\big)^{-1}\) for each \(\xi>0,\) where \(-\Delta\) is the Neumann-Laplacian in \(L^2(\Omega).\) It is worth noting that in the one-dimensional case \((N=1),\) \(\big(\xi+n^{-1}(-\Delta)+T_p\big)^{-1}\) converges strongly to \((\xi+T_p)^{-1}\) in \(L^p[0,1]\) with \(T_p\) being the one-dimensional analog constructed in [M. Campiti et al., Semigroup Forum 57, No. 1, 1–36 (1998; Zbl 0915.47029)].