Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?

The boundary condition is that the solution at the boundary is zero. Assume that $D$ is a unit ball in $R^{d}$. Since the eigenvalues of this operator are countable and nonnegative, the spectral gap is the difference between its smallest nonzero eigenvalue and zero.