Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

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 Panetiz-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.

Bounds to the first eigenvalues of weighted p-Steklov and (p,q)-Laplacian Steklov problems

“…Now, assume that f is not constant. If equality occurs, then the end of the proof is similar to the proof of Roth [15].…”

“…T e i , e j B(e i , e j ), where {e 1 , • • • , e n } is a local orthonormal frame of T ∂M and B is the second fundamental form of the immersion of M into R N . We also, recall the generalized Hsiung-Minkowski formula [11,14,15] as…”

“…For codimension greater than 1, H r is a function and H r+1 is a normal vector field. These inequalities have been generalized for other ambient spaces and other operators (see [1,2,3,7,8,11,13,14,15,16]). Du and Mao [9] established the first positive eigenvalue of the p-Laplacian on closed submanifold of R N satisfies the follows inequalities.…”

Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

“…Some of the inequalities for eigenvalues presented above are included in our results, but before presenting our theorems, we first denote by H T the generalized mean curvature vector associated with (1, 1)-tensor T , see Section 2 for more details. The definition of the generalized mean curvature vector was considered by Grosjean [14] and Roth [22]. Moreover, since the (1, 1)-tensor T is symmetric and positive definite and Ω is a bounded domain in Problem 1.2, there are positive real numbers ε and δ such that εI ≤ T ≤ δI where I is the identity (1, 1)-tensor.…”

Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds

“…On the other hand, more recently, in [14], the second author prove the following general inequality (7)…”