SHEELA VERMA
Articles written in Proceedings – Mathematical Sciences
Volume 130 All articles Published: 13 February 2020 Article ID 0021 Research Article
Upper bound for the first nonzero eigenvalue related to the $p$-Laplacian
Let $M$ be a closed hypersurface in $\mathbb{R}^{n}$ and $\Omega$ be a bounded domain such that $M = \partial\Omega$. In this article, we obtain an upper bound for the first nonzero eigenvalue of the following problems:
(1)
$$\Delta_{p}u = \lambda_{p} |u|^{p−2} u \quad {\rm on}\; M.$$
(2)
$$\begin{array}{ll} \Delta_{p}u =0 \quad \;\;\;\;\;\;\rm{in} \;\Omega,\\ |\nabla u|^{p−2} \frac{\partial u}{\partial v} = \mu_{p}|u|^{p−2} u \quad {\rm on} \ M. \end{array}$$
These bounds are given in terms of the first nonzero eigenvalue of the usual Laplacianon the geodesic ball of the same volume as of $\Omega$.
Volume 130, 2020
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