I need upper bounds for the norm in L^2 of a function u such that -\Delta u=\lambda u on \Omega, u vanishes on the boundary. If the set \Omega is a polyhedron we have that this norm equals(!) the L^2-norm of u multiplied by the p/2 -power of \lambda; p is the order of the derivative. It is not true for the 3-dimensional ball and eigenfunction u=\sin r/r but numerical experiments suggest that the equality can be repaced by the inequality

|| D^p u || \le \lambda ^{p/2} ||u||.

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