Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization (Q998725)

The paper deals with the fixed membrane problem \[ \Delta u+\lambda u=0 \quad\text{in } D, \qquad u=0 \quad\text{on }\partial D, \] where \(D\) is a bounded domain of \(\mathbb R^n\). It is well-known that Schwarz symmetrization allowed to obtain estimates for the first eigenfunction and for the eigenvalues \(\lambda_i\) of the problem under consideration. \textit{G. Chiti} [Boll. Unione Mat. Ital., VI. Ser., A 1, 145--151 (1982; Zbl 0484.35067)] proved pointwise estimations for the Schwarz rearrangement of \(u\). Basing on Chiti's result the author gives estimates for the first eigenfunction which are sharper than the results given in the above mentioned paper. The paper contains also an extension to a class of more general elliptic problems.