Sharp Steklov upper bound for submanifolds of revolution (Q2238548)

The authors consider the classical Steklov problem for the Laplace-Beltrami operator defined on the boundary of a \(n\)-dimensional submanifold of revolution \(M\) embedded in \(\mathbb R^{n+1}\): \[ \begin{cases} -\Delta u =0, & \text{in } M,\\ \partial_\nu u=\sigma u, & \text{on }\Sigma, \end{cases} \] where \(\Sigma\) is the boundary of \(M\) and \(\nu\) is the canonical exterior normal to \(\Sigma\) in \(M\). We recall that this Steklov problem admits a sequence of nonnegative eigenvalues of finite multiplicities diverging to plus infinity, and enjoying a Courant minmax characterization. If we consider \(\sigma_{(k)}(M)\), the \(k\)-th distinct eigenvalue, that is without considering multiplicities, then \[ 0=\sigma_{(0)}(M)<\sigma_{(1)}(M)<\sigma_{(2)}(M)<\dots \] The main theorem states that, for \(n\ge 3\) and \(k\ge 1\), for any given manifold of revolution \(M^n\subset\mathbb R^{n+1}\) with one boundary component \(\Sigma\) isometric to \(\mathbb S^{n-1}\), then \[ \sigma_{(k)}(M)<k+n-2, \] and moreover this bound is sharp. The proof is based on a nice Dirichlet-Neumann bracketing technique on spherical shells where explicit computations are feasable and immediately show the general behaviour. This result complements those in [\textit{B. Colbois} et al., Can. Math. Bull. 63, No. 1, 46--57 (2020; Zbl 1433.35205)] where lower bounds are proved.