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

We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem ”u+ u=0 in D ￠ Rn 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>4p ￠ ’1. On the other hand, if D ￠ R2, resp. R3 the norm ￠ D|gradu|2dx=1 implies max|uk| ￠ ’k ￠ ’ ￠ 0, resp. max|uk|=0(1).