• SHEELA VERMA

      Articles written in Proceedings – Mathematical Sciences

    • Upper bound for the first nonzero eigenvalue related to the $p$-Laplacian

      SHEELA VERMA

      More Details Abstract Fulltext PDF

      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) Closed eigenvalue problem:

      $$\Delta_{p}u = \lambda_{p} |u|^{p−2} u \quad {\rm on}\; M.$$

      (2) Steklov eigenvalue problem:

      $$\begin{array}{ll} \Delta_{p}u =0 \quad \;\;\;\;\;\;\rm{in} \;\Omega,\\ |\nabla u|^{p−2} \frac{\partial u}{\partial v} = \mu_{p}|u|^{p−2} u \quad {\rm on} \ M. \end{array}$$

      These bounds are given in terms of the first nonzero eigenvalue of the usual Laplacianon the geodesic ball of the same volume as of $\Omega$.

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