We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^{-1,\alpha }(\mathrm{\Omega })$ consinsting of all derivatives of hölder-continuous functions in $\mathrm{\Omega }$ where $\mathrm{\Omega }$ is a domain in ${ℝ}^n$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^{-1,\alpha }(\mathrm{\Omega })$. We prove also that the spaces $C^{-1,\alpha }(\mathrm{\Omega })$ can be considered as extrapolation spaces relative to suitable non-variational operators....

We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations ${\partial }_t\left(r_{h}u\right)-\mathrm{div}(a_h·Du)$ with $r_h(x,t)\ge 0$, $r_h\in L^{\infty }(\Omega \times (0,T))$. The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators $(r_h\equiv 0)$, G-convergence for parabolic operators $(r_h\equiv 1)$, singular perturbations of an elliptic operator $(a_h\equiv a$ and $r_h\to r$, possibly $r\equiv 0)$.