Weinstock inequality in higher dimensions. (English) Zbl 1468.35038
Summary: We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in \(\mathbb{R}^n\), for \(n \geq 3\), in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 47J30 | Variational methods involving nonlinear operators |
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