On the Eigenvalues of the Polyharmonic Operator

Let B be the unit ball of the n-dimensional Euclidean space. We study the eigenvalue problem associated with the polyharmonic operator in B. We prove that the first eigenvalue is positive and simple, and is the unique eigenvalue which admits eigenfunctions of constant sign. We prove some regularity properties of the first eigenfunction. Moreover, we investigate the variational characterization of the first eigenvalue. We give some sufficient conditions for radiality and monotonicity of the first eigenfunction. We use the tools of the Green function and the Krein-Rutman theorem.