Subharmonicity, comparison results, and temperature gaps in cylindrical domains. (Q265212)

Let \(D\subset \mathbb R^{n-1}\) be a bounded domain and \(\Omega =D\times (0,l)\) be a cylinder in \(\mathbb R^n\). For \(u\) defined in \(\Omega \) let \(u^*\) be the function introduced by Baernstein. It is proved for \(u(x,t)\in C^2(\overline \Omega)\) with \(-\Delta u=f\) in \(\Omega \), \(u_t(x,0)=u_t(x,l)=0\) that \(-\Delta u^* \leq f^*\). In particular, if \(u\) is subharmonic then \(u^*\) is weakly subharmonic. Then the comparison theorem for the Neumann problem of the Poisson equation in cylinders is proved.