Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

Abstract

We establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

The Cheeger problem in abstract measure spaces

“…Here, 𝐸 (𝑡) denotes the set of points of density 𝑡 ⩾ 0 for 𝐸 and we recall that ℒ 𝑛 (ℝ 𝑛 ⧵ (𝐸 (0) ∪ 𝐸 (1) )) = 0, see, for example, [98,Thm. 16.3] for more details.…”

“…In Theorem 3.6, we provide a general existence result. Unsurprisingly, the key assumptions on the perimeter are the lower semicontinuity and the compactness of its sublevels with respect to the 𝐿 1 norm, besides an isoperimetric-type property that prevents minimizing sequences to converge toward sets with null 𝔪-measure.…”

“…Let (𝑋, 𝒜, 𝔪) be a measure space, and let Ω ∈ 𝒜. A function 𝐻 ∈ 𝐿1 (Ω, 𝔪) is said to be a 𝑃-mean curvature in Ω of a set 𝐸 ⊂ Ω if 𝐸 minimizes the functional…”

The Cheeger problem in abstract measure spaces

“…Here, E (t) denotes the set of points of density t ≥ 0 for E and we recall that L n (R n \(E (0) ∪E (1) )) = 0, see, e.g., [93,Thm. 16.3] for more details.…”

“…Since then, the Euclidean case has been the most studied, and we refer to the two surveys [87,108], but many different contexts have been object of research, such as: weighted Euclidean spaces [7,26,91,111]; finite dimensional Gaussian space [50,81]; anisotropic Euclidean and Riemannian spaces [8,22,48,83]; the fractional perimeter [28] or non-singular non-local perimeter functionals [97]; Carnot groups [104]; RCD-spaces [67,68]; and lately smooth metric-measure spaces [1].…”