Eigenvalue inequalities for the p -Laplacian on a Riemannian manifold and estimates for the heat kernel

Abstract

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirich-

The Gagliardo-Nirenberg Inequalities and Manifolds With Non-Negative Weighted Ricci Curvature

“…As shown in [14] and also pointed out in [24], the facts (2.5) and (2.6) have a fundamental role in the derivation of the generalized Bishop's volume comparison theorem I below (see Theorem 2.5 for the precise statement). One can also find that (2.6) is also necessary in the proof of Theorem 1.3 in Section 3.…”

“…Clearly, v z is in the radial direction. In fact, the notion of having radial curvature bound has been used by the author in [14,24,25] to investigate some problems like eigenvalue comparisons for the Laplace and p-Laplace operators (between the given complete manifold and its model manifold), the heat kernel comparison, etc. This notion can also be found in other literature (see, for instance, [19,29]).…”

Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates

“…This is because for any ξ ∈ S n−1 p and t 0 > 0, we have ∇r(γ ξ (t 0 )) = γ ′ ξ (t 0 ) when the point γ ξ (t 0 ) = exp p (t 0 ξ ) is away from the cut locus of p. We need the following concepts. [16,20,21]) Given a continuous function k : [0, l) → R, we say that M has a radial Ricci curvature lower bound (n − 1)k along any unit-speed minimizing geodesic starting from a point p ∈ M if…”

“…where Ric is the Ricci curvature of M. 16,20,21]) Given a continuous function k : [0, l) → R, we say that M has a radial sectional curvature upper bound k along any unit-speed minimizing geodesic starting from a point p ∈ M if…”

“…This inequality makes an important role in the study of existence and regularity of solutions of some boundary value problems. In order to state our main conclusions below clearly, here we would like to introduce some basic notions, which have been introduced in [16,20,21,22] already. Besides, in the sequel, for convenience, we will drop the integral measures for all integrals except it is necessary.…”