- 期刊
EQUATION OF THE FIXED MEMBRANE IN THE N-DIMENSIONAL SPACE: SOME REMARKS ON THE MAXIMA OF THE EIGENFUNCTIONS SUBJECTED TO VARIOUS NORMS
並列摘要
We show in this paper that the sequence {max |u_k|}, where the u_k are the elgenfunctlons of the problem ∆u + λu= 0 in D c R^n and u = 0 on □D, is not bounded generally if one imposes the norm ʃ_D u^(2p)(x)dx= 1, p =(1),2,3… The same holds with the norm ʃ_D |gradu|^(2P) dx = 1 when n ＞ 4p-1 On the other hand, if D c R^2 resp. R^3 the norm ʃ_D |gradu|^2dx = 1 implies max|u_k| →_(K→∞) 0, resp. max |u_k|= 0(1).
延伸閱讀
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