Inequalities between Dirichlet and Neumann eigenvalues for domains in spheres.
Let \(M\) be a compact Riemannian manifold with smooth boundary \(\partial M\) and let \(0<\lambda_1<\lambda_2 \leq \lambda_3 \leq \cdots\) and \(0=\mu_0<\mu_1 \leq \mu_2 \leq \cdots\) be the Dirichlet and Neumann eigenvalues, respectively, of the Laplace-Beltrami operator on \(M\). Many relations between the Dirichlet and Neumann eigenvalues are known. For example, the variational characterization of the eigenvalues implies that \(\mu_{k-1}\leq \lambda_k\) for \(k=1,2,\cdots\). For compact subdomains of Euclidean space, \textit{L.~Friedlander} [Arch. Rat. Mech. Anal. 116, No. 2, 153--160 (1991; Zbl 0789.35124)] proved that \(\mu_k\leq \lambda_k\) for \(k=1,2,\cdots\). This result was extended to compact subdomains of symmetric spaces of noncompact type by \textit{R.~Mazzeo} [Int. Math. Res. Not. 1991, No. 4, 41--48 (1991; Zbl 0752.58035)], but it is false in general. Indeed, \(\mu_1>\lambda_1\) for a geodesic ball \(B(r)\) of radius \(r\) in the unit sphere \({\mathbf S}^n\) if \(B(r)\) contains a hemisphere. Here the authors study the case of subdomains of \({\mathbf S}^n\). Adopting the sign convention that the second fundamental form of the boundary of the geodesic ball \(B(r)\) has second fundamental form \(-(\cot r)\delta_{ij}\), they prove the following. If \(M\) is a subdomain of \({\mathbf S}^n\) with smooth boundary \(\partial M\) of nonpositive mean curvature, then \(\mu_k\leq \lambda_k\) for \(k=1,2,\cdots\). Moreover, if equality holds for some \(k\), then \(\partial M\) is a minimal hypersurface. A second result is the following. If \(M\) is as above, but now the second fundamental form of \(\partial M\) is nonpositive, then \(\mu_{k+n_0}\leq \lambda_k\) for \(k=1,2,\cdots\) where \(n_0=\left[{{n-1}\over 2}\right]\), the greatest integer less than or equal to \({{n-1}\over 2}\). The proof uses a technique of \textit{H. A.~Levine} and \textit{H. F. Weinberger} [Arch. Rat. Mech. Anal. 94, 193--208 (1986; Zbl 0608.35047)] for constructing suitable test functions.
scientific article; zbMATH DE number 1743678
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