Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space (Q2421243)

The authors consider upper and lower bounds for the eigenvalues of the Steklov problem \[ \begin{cases} \Delta u=0, & \text{ in }M\\ \partial_{\nu}u=\sigma u, & \text{ on }\Sigma \end{cases} \] where \(\Sigma\) is a \(n-1\)-dimensional compact smooth submanifold in \(\mathbb R^m\) and \(M\) is a \(n\)-dimensional compact connected smooth submanifold of \(\mathbb R^m\) with boundary \(\Sigma\), and \(\partial_{\nu}u\) is the outward-pointing normal derivative on \(\Sigma\). The spectrum is discrete and is given by an increasing sequence of non-negative eigenvalues of finite multiplicity \[ 0=\sigma_1<\sigma_2\leq\cdots\leq\sigma_k\leq\cdots\nearrow+\infty \] An upper bound on the eigenvalues is given in terms of the volume of \(M\). Namely, the authors prove that there exists a constant \(A_{\Sigma}\) depending on \(\Sigma\) such that any \(n\)-dimensional submanifold \(M\) of \(\mathbb R^m\) with boundary \(\Sigma\) satisfies \[ \sigma_k\leq A_{\Sigma}|M|k^{\frac{2}{n-1}}. \] The constant \(A_{\Sigma}\) depends on the geometry of \(\Sigma\) via a lower bound on its Ricci curvature, an upper bound on the diameter, the number of connected components of \(\Sigma\) and their maximal distorsion. The distorsion of a connected submanifold \(N\) of \(\mathbb R^m\) is defined as \(\sup_{x,y\in N, x\ne y}\frac{d_N(x,y)}{|x-y|}\), where \(d_N\) is the geodesic distance on \(N\). Moreover, if \(\Omega\) is a domain in \(\mathbb R^n\), considered as a linear subspace of \(\mathbb R^m\), with boundary \(\Sigma\) then \[ \sigma_k\leq A(n)\frac{|M|}{|\Omega|^{\frac{n+1}{n}}}k^{\frac{2}{n-1}}, \] for any \(n\)-dimensional submanifold \(M\) of \(\mathbb R^m\) with boundary \(\Sigma\). As for lower bounds, the authors prove that any hypersurface of revolution \(M\subset\mathbb R^{n+1}\), \(n\geq 3\), with bounrday \(\mathbb S^{n-1}\times\{0\}\) satisfies \(\sigma_k\geq\sigma_k(\mathbb B^n)\) with equality if and only if \(M=\mathbb B^n\times\{0\}\). The authors also prove an analogous lower bound for hypersurfaces of revolution with two boundary components.