2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218505We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $Ω_1$ of the domain of integration of $Ω$. We generalize Lions' results to covering the case of diffusivities which are piecewise $C^1$ up to the boundary of $Ω_1$ and $Ω_2$, where $Ω_2 := Ω\setminus \overlineΩ_1$ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of $(A_{\bar{p}_ν}^{-1})_{ν\in \mathbb{N}^\ast},$ for a monotonically decreasing sequence of diffusivities $(\bar{p}_ν)_{ν\in \mathbb{N}^\ast}$.Analysis of PDEsFunctional Analysis35J25, 47F05, 65J10, 65N99text