Estimates for the first eigenvalues of divergence type operators (Q6545062)

Let \((M,g)\) be an \(n\)-dimensional compact Riemannian manifold with boundary \(\partial M\). Let \(\phi\) and \(V\) be two smooth and positive functions defined on \(M\). We consider \(M\) endowed with the weighted measure \(d\mu=e^{-\phi}dv_g\).\N\NThe central object of the paper is the linear elliptic operator \(L^g_{\phi,V}\) defined as \N\[\NL^g_{\phi,V}=e^\phi\mathrm{div}\left[e^{-\phi}V^2\nabla\Big(\frac{\cdot}{V}\Big)\right],\N\]\Nwhose study is motivated by the Reilly type formulas obtained in [\textit{G. Huang} et al., Differ. Geom. Appl. 94, Article ID 102136, 21 p. (2024; Zbl 1540.53062)]. The present paper provides lower bounds on the first nonzero eigenvalue of the square of the operator \(L^g_{\phi,V}\) for various eigenvalue problems, depending on the imposed boundary conditions, assuming appropriate lower bounds on the modified weighted Ricci curvature \N\[\N\widehat{\mathrm{Ric}}^V_{\phi,m}:=\frac{\Delta_\phi V}{V}g-\frac{1}{V}\nabla^2V+\mathrm{Ric}_{\phi,m}, \qquad m\in \mathbb{R}\cup{\infty}.\N\]\NNotice that when \(V\equiv 1\) we have \(L^g_{\phi,V}=\Delta_\phi\) and \(\widehat{\mathrm{Ric}}^V_{\phi,m}=\mathrm{Ric}_{\phi,m}\), where \(\Delta_\phi\) is the drifted Laplacian (also known as Witten Laplacian) and \(\mathrm{Ric}_{\phi,m}\) is the \(m\)-dimensional Bakry-Émery Ricci curvature. In this situation, the results of the article recover some eigenvalue estimates previously obtained in the most classical setting of a bi-drifted Laplacian \(\Delta_\phi^2=\Delta_\phi(\Delta_\phi)\), so the novelty of the paper lies in considering a general (positive) function \(V\).