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International Journal of Mathematics and Mathematical Sciences
Volume 2, Issue 2, Pages 337-340
http://dx.doi.org/10.1155/S0161171279000296
Research notes

Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms

Department of Mathematics, Cornell University, Ithaca 14850, New York, USA

Received 15 March 1979

Copyright © 1979 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in DRn and u=0 on D, is not bounded generally if one imposes the norm Du2p(x)dx=1, p=(1),2,3,. The same holds with the norm D|gradu|2pdx=1 when n>4p1. On the other hand, if DR2, resp. R3 the norm D|gradu|2dx=1 implies max|uk|k0, resp. max|uk|=0(1).