Isoperimetric Upper Bounds and Reilly-Type Inequalities for the First Eigenvalue of the $p$-Biharmonic Operator

Abstract:

      In this paper, we give some isoperimetric upper bounds for the first eigenvalue of the $p$-biharmonic operator of an $n$-dimensional embedded closed hypersurface in an Euclidean space. We also give Reilly-type inequalities for the first eigenvalue of the $p$-biharmonic operator of an $n$-dimensional closed submanifold immersed into a higher dimensional manifold such as an Euclidean space, a unit sphere, a projective space.