Abstract
Let $M$ be a domain in the unit $n$-sphere with smooth boundary. The purpose of this paper is to describe some inequalities between Dirichlet and Neumann eigenvalues for $M$ under certain convex restrictions on the boundary. We prove that if the mean curvature of the boundary is nonpositive, then the $k$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1, 2,\cdots$. Furthermore, if the second fundamental form of the boundary is nonpositive, then the $(k+\left[\frac{n-1}{2}\right])$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1,2,\cdots$.
Citation
Yi-Jung Hsu. Tai-Ho Wang. "INEQUALITIES BETWEEN DIRICHLET AND NEUMANN EIGENVALUES FOR DOMAINS IN SPHERES." Taiwanese J. Math. 5 (4) 755 - 766, 2001. https://doi.org/10.11650/twjm/1500574993
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