On the L∞ Norm of the First Eigenfunction of the Dirichlet Laplacian期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the L∞ Norm of the First Eigenfunction of the Dirichlet Laplacian

Authors:	van den Berg M

Institution:	(1) School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, England

Abstract:	We obtain the asymptotic behaviour for the L norm of the first eigenfunction of the Dirichlet Laplace operator on a conic sector over a geodesic disc ${B_{\eta} }$ in $\mathbb{S}^{m - 1}$ as ${\eta} \to {0}$ . We are led to conjecture that for an open, bounded and convex set D with inradius and diameter d, $\left\\| \phi \right\\|_\infty \leqslant c_m \rho ^{{{\left( {1 - 3m} \right)} \mathord{\left/ {\vphantom {{\left( {1 - 3m} \right)} 6}} \right. \kern-\nulldelimiterspace} 6}} d^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 6}} \right. \kern-\nulldelimiterspace} 6}}$ where $\left\\| \phi \right\\|_2$ and

Keywords:	Dirichlet Laplacian eigenfunction
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