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 | u k | } , where the u k are the eigenfunctions of the problem Δ u + λ u = 0 in D ⊂ R n and u = 0 on ∂ D , is not bounded generally if one imposes the norm ∫ D u 2 p ( x ) d x = 1 , p = ( 1 ) , 2 , 3 , … . The same holds with the norm ∫ D | gradu | 2 p d x = 1 when n > 4 p − 1 . On the other hand, if D ⊂ R 2 , resp. R 3 the norm ∫ D | gradu | 2 d x = 1 implies max | u k | → k → ∞ 0 , resp. max | u k | = 0 ( 1 ) .