On sign preservation for clotheslines, curtain rods, elastic membranes and thin plates (Q343406)

This is a well-written survey concerning positive solutions to boundary value problems for second and fourth order elliptic differential equations. Let \(\Omega \subset \mathbb R^n\) be a bounded smooth domain and \(f\) a given positive function in \(\Omega\). One of the various problems can be stated for a function \(u\) satisfying equation \(\Delta^2u +\lambda u=f\) in \(\Omega\) and the boundary conditions \(u=0\) and \(-\Delta u= -(1-\sigma) \kappa\; \frac{\partial u}{\partial \nu}\) on \(\partial \Omega\) where \(\frac{\partial}{\partial \nu}\) denotes the outward normal derivative and \(\kappa\) the signed curvature of the boundary, positive on convex boundary sections and negative on concave parts; the constant \(\sigma \in (-1,1)\). The main result concerns estimations of \(\lambda\) for which positive solutions exist. The Krein-Rutman theorem devoted to positive operators is used in the study. Special attention is payed to the one-dimensional case. The case of corner points on \(\partial \Omega\) is discussed in Sobolev's space.