Estimates for eigenfunctions of elliptic operators with respect to the spectral parameter (Q1966304)

The paper deals with the uniformly estimates for the \(L_2\)-normalized eigenfunctions \(u_n(x)\) of the spectral boundary value problem \[ Lu\equiv \sum^N_{i,j= 1} {\partial\over\partial x_i} \Biggl(a_{ij}(x){\partial u\over\partial x_i}\Biggr)+ a(x) u+\lambda u= 0,\quad u|_{\partial D}= 0, \] with measurable coefficients in a domain \(D\subset \mathbb{R}^N\). The main result is \[ \max_{\overline D}|u_n(x)|\leq C\lambda^{N/2}_n, \] where \(C\) is constant depending only on \(N\) and \(a_0\). Examples of operators for which this estimate is attained are also given.