Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues
- PDF / 481,384 Bytes
- 26 Pages / 439.642 x 666.49 pts Page_size
- 58 Downloads / 172 Views
Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues Julien Roth1 Received: 21 June 2017 / Accepted: 27 May 2019 / © Springer Nature B.V. 2019
Abstract We prove Reilly-type upper bounds for different types of eigenvalue problems on submanifolds of Euclidean spaces with density. This includes the eigenvalues of Paneitz-like operators as well as three types of generalized Steklov problems. In the case without density, the equality cases are discussed and we prove some stability results for hypersurfaces which derive from a general pinching result about the moment of inertia. Keywords Paneitz operator · Steklov problem · Eigenvalues · Hypersurfaces · Pinching Mathematics Subject Classification (2010) 35P15 · 53C20 · 53C24 · 53C42 · 58C40
1 Introduction Let (M n , g) be a n-dimensional (n 2) closed, connected, oriented manifold, isometrically immersed by X into the (n + 1)-dimensional Euclidean space Rn+1 . The spectrum of the Laplacian of (M, g) is an increasing sequence of real eigenvalues 0 = λ0 (M) < λ1 (M) λ2 (M) · · · λk (M) · · · −→ +∞, counted without multiplicity. The eigenvalue 0 (corresponding to constant functions) is simple and λ1 (M) is the first positive eigenvalue. In [30], Reilly proved the following well-known upper bound for λ1 (M) n λ1 (M) H 2 dvg , (1) V (M) M where H is the mean curvature of the immersion. In the same paper, he also proved an analogous inequality involving the higher order mean curvatures. Namely, for r ∈ {1, · · · , n} 2 Hr−1 dvg V (M) Hr2 dvg , (2) λ1 (M) M
M
Julien Roth
[email protected] 1
Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UPEM-UPEC, CNRS, F-77454 Marne-la-Vall´ee, France
J. Roth
where Hk is the k-th mean curvature, defined as the k-th symmetric polynomial of the principal curvatures. Moreover, Reilly studied the equality cases and proved that equality in Eq. 1 as in Eq. 2 is attained if and only if X(M) is a geodesic sphere. These inequalities have been generalized for other ambient spaces [19, 21], other operators, in particular of Jacobi type [1, 4, 7], in the anisotropic setting [33] or for weighted ambient spaces [9, 18, 34]. In particular, in [34], we prove the following general inequality
2 tr (S)μf
λ1 (LT ,f ) M
tr (T )μf M
HS 2 + S∇f 2 μf ,
(3)
M
where μf = e−f dvg is the weighted measure of (M, g) endowed with the density e−f , T , S are two symmetric, free-divergence (1, 1)-tensors with T positive definite. Moreover, LT ,f is the second order differential operator defined for any smooth function u on M by LT ,f = −div(T ∇u) + ∇f, T ∇u. In this paper, we are interested in a closely related problems. First, we prove eigenvalue estimates for the Steklov problem for submanifolds with boundary. We obtain general Reilly-type upper bounds (see Theorem 3.1) that generalize the earlier estimates by Ilias and Makhoul [26]. These upper bounds are given for a larger class of a general Steklov problem, also involving a weighted measure. We also consider two other Steklov-type problems
Data Loading...
