EXTRINSIC UPPER BOUNDS THE FIRST EIGENVALUE OF THE p-STEKLOV PROBLEM ON SUBMANIFOLDS

Abstract

We prove Reilly-type upper bounds for the first non-zero eigen-value of the Steklov problem associated with the p-Laplace operator on sub-manifolds with boundary of Euclidean spaces as well as for Riemannian products R × M where M is a complete Riemannian manifold.

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

“…First, assume that f is constant, H does not vanish identically, and equality holds. Then the end of the proof is similar to the proof of Roth [16] for the p-Steklov problem. Now, assume that f is not constant.…”

“…Proof of theorem 2.4. Similar to [16], we assume that the function t is a test function. Let v = ∂ t , ν = ∇t, ν .…”

“…In following we will consider the weighted p-Steklov problem on submanifolds with boundary of the Euclidean space (1.1) ∆ p,f u = 0 in M |∇u| p−2 ∂u ∂ν = λ|u| p−2 u on ∂M, where ∂u ∂ν is the derivative of the function u with respect to the outward unit normal ν to the boundary ∂M . If f be a constant function then the weighted p-Steklov problem (1.1) reduces to the p-Steklov problem which it has been studied in [16]. This problem arises from the following variational characterization of the first positive eigenvalue given by (1. where dµ h is the weighted measure on ∂M .…”

“…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.…”

“…In addition, equality holds if and only if p = 2 and M is minimally immersed into a geodesic hypersphere. On the other hand, Roth [16] proved Reilly-type inequalities for the first eigenvalue of p-Steklov problem on submanifolds of R N and showed that…”