Upper Bounds on the First Eigenvalue for the p -Laplacian
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Upper Bounds on the First Eigenvalue for the p-Laplacian Zhi Li and Guangyue Huang Abstract. In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the p-Laplacian Δp u = −λ|u|p−2 u with p > 1 on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the p-Laplacian. Mathematics Subject Classification. 58J05, 35J92. Keywords. Eigenvalue, p-Laplacian, gradient estimates.
1. Introduction It is well known that using a comparison theorem for the first Dirichlet eigenvalue of the Laplacian of a geodesic ball on a complete Riemannian manifold with Ricci curvature bounded from below, Cheng [2] derived an estimate for the bottom of L2 -spectrum on such manifolds by letting the radius of the ball to infinity. In particular, if the Ricci curvature bounded from below by −(n − 1)K 2 for a constant K, Cheng obtained the following upper bound estimate(see Theorem 4.2 in [2]) on the first nontrivial eigenvalue of the Laplacian: (n − 1)2 K 2 . (1.1) 4 As we know, using the maximum principle to deal with gradient estimates on Laplacian equations is a power tool in geometric analysis. For example, Yau [20] showed that every positive or bounded solution to the heat equation is constant. Li [6] derived several Liouville type theorems with respect to the weighted Laplacian. Generalizing Theorem 4.2 of Cheng [2], Wang [16] and Wu [19] obtained upper bound estimates of weighted Laplacian with ´ respect to the Bakry–Emery Ricci curvature, respectively. λ1 (Δ) ≤
Research supported by NSFC (Nos. 11971153, 11671121).
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It is a natural question to derive upper bound estimates of the first eigenvalue of the p-Laplacian. The p-Laplacian on an n-dimensional Riemannian manifold (M n , g) is defined by Δp u := div(|∇u|p−2 ∇u),
for u ∈ W 1,p (M n ),
where the equality is in the weak W 1,p (M n ) sense (p > 1), which can be seen as a generalization of the Laplacian. In fact, the so-called p-Laplacian eigenvalue problem is to consider the following nonlinear second-order partial differential equation: Δp u = −λ|u|p−2 u in the distribution sense and u is said to be an eigenfunction associated with λ. By standard regularity results for non-uniformly elliptic equations, one has that the regular weak solutions u ∈ C 1,α (M n ) ∩ W 1,p (M n ) for some 0 < α < 1 and u is a smooth function where ∇u = 0 and u = 0, then u ∈ C 3,α (U ) if p > 2 and u ∈ C 2,α (U ) for 1 < p < 2, where U is a suitably small neighborhood of x ∈ M n . The standard reference for these results is [11], where the problem is studied in local coordinates. Regarding regularity issues, it is worth mentioning the very recent article [9], which obtains studies the behaviour of the exponent at points where ∇u = 0. Besides, interested readers can find the classical papers by K. Uhlenbeck [12] and L. Evans [4]. Now, we mainly study the infimum of the positive eigenvalues as the first nonzero eigenvalue or si
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