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

[摘要] We show in this paper that the sequence{max|uk|}, where theukare the eigenfunctions of the problemΔu+λu=0inD⊂Rnandu=0on∂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=1whenn>4p−1. On the other hand, ifD⊂R2, resp.R3the norm∫D|gradu|2dx=1impliesmax|uk|→k→∞0, resp.max|uk|=0(1).

[学科分类] 数学（综合）