On the Spectra of Three Steklov Eigenvalue Problems on Warped Product Manifolds

Let $$M^n=[0,R]\times \mathbb {S}^{n-1}$$ be an n-dimensional ( $$n\ge 2$$ ) smooth Riemannian manifold equipped with the warped product metric $$g=dr^2+h^2(r)g_{\mathbb {S}^{n-1}}$$ and diffeomorphic to a Euclidean ball. Assume that M has strictly convex boundary. First, for classical Steklov eigenvalue problem, we obtain optimal lower (upper, respectively) bound its spectrum in terms of $$h'(R)/h(R)$$ when $$\mathrm {Ric}_g\ge 0$$ $$\le , respectively). Second, two fourth-order problems studied by Kuttler Sigillito 1968, derive their spectra either $$h'(R)/h^3(R)$$ or which is certain cases; particular, confirm conjecture raised Q. Wang C. Xia manifolds dimension $$n=2$$ 4$$ . For some proofs utilize Reilly’s formula reveal new feature on use.