Consider a periodic elliptic operator $L=\partial_{x_i}(a^{ij}(x)\partial_{x_j})$ on the torus say, satisfying $a^{ij}(x)\xi_i \xi_j \geq \nu |\xi|^2$ for all $x \in \mathbb{T}^d$, but further satisfying $a^{ij}(x)\xi_i \xi_j \geq (\nu+\alpha) |\xi|^2$ say for some $\alpha>0$ for all $x \in A \subset \mathbb{T}^d$, where $A$ is some set of positive Lebesgue measure.
If one wishes to consider the spectrum of said operator on the space of mean zero functions, a naive application of the Poincare inequality would suggest that the smallest eigenvalue is bounded below by $$ \lambda_{min} \geq C^{-1}_{poincare}\nu $$ where $C_{poincare}$ is the smallest constant so that $||u||_{L^2}^2\leq C_{poincare}||Du||_{L^2}^2$ holds for all mean zero functions $u$.
However, this doesn't seem sharp as we are essentially "throwing away" the stronger ellipticity we have in the set $A$ (although I nevertheless suspect there are situations where one cannot do better).
I was wondering if there are any known results pertaining to situations in which this added ellipticity does indeed yield stronger lower bounds on the first eigenvalue.