Lower estimates on the principal eigenvalue of Witten q-Laplacian on Smooth metric measure spaces

Some eigenvalue inequalities in terms of generalized Mckean bound and the weighted Cheeger’s constant are proved for Witten q-Laplacian on smooth metric measure spaces with smooth boundary imposing certain restrictions on geometric quantities such as mean curvatute and sectional curvature of the domains. On the other hand, a clamped plate problem involving Witten bi-Laplacian is considered on a weighted manifold in the regime of positive generalized Ricci curvature, while lower bound estimates on its principal frequency are established. Indeed, the principal frequency of the problem is shown to be bounded from below by a double of McKean bound, provided the generalized curvature is nonnegative. As an application clamped plate eigenvalue lower bound is derived on the weighted geodesic ball having nonnegative generalized Ricci curvature.