Some bounds for eigenfunctions of an elliptic operator (Q1073268)

The authors study an elliptic problem \[ Lu=\lambda^{2m}u\quad (in \Omega),\quad B_ ju|_{\partial \Omega}=0\quad (j=1,...,m);\quad \| u\|_{L^ 2(\Omega)}=1, \] here \(\Omega \subset {\mathbb{R}}^ n\), \(\partial \Omega\) and the coefficients are smooth. The estimate (1) \(\max_{x\in {\bar \Omega}}| u(x)|^ 2\leq c| \lambda^ n|\) is proved; the constant \(c>0\) does not depend on \(\lambda\). In the case of Dirichlet problem for the equation with constant coefficients under certain conditions the estimate \(\max_{x\in {\bar \Omega}}| u(x)|^ 2\leq c| \lambda^{n-1}| \ell n| \lambda |\) is proved. The estimate (1) is proved also for the second order equation with non-smooth coefficients.