Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space R ⁿ with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in R ⁿ .