ISSN:
1572-929X
Keywords:
Dirichlet
;
Laplacian
;
eigenfunction
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We obtain the asymptotic behaviour for the L ∞ norm of the first eigenfunction φ 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 ρ 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 $$c_m$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1026452623177
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