Steiner symmetrization and periodic solutions of boundary value problems (Q1332670)

Summary: Let \(f^* = f^*(x,y)\) denote the monotone decreasing rearrangement of a function \(f = f(x,y)\) with respect to \(y\). If \(-\Delta u = f\), \(-\Delta v = f^*\) in the domain \(\Omega = (0,1) \times (0,1)\) and \(\partial u/ \partial n = \partial v/ \partial n = 0\) on the boundary \(\partial \Omega\) of \(\Omega\), then \(\text{osc} u \leq \text{osc} v\), where the quantity osc \(w\) for a function \(w\) is defined as the difference \(\sup w - \inf w\). Similar results are proved for periodic solutions of some boundary value problems in cylindrical domains.