Subharmonicity, comparison results, and temperature gaps in cylindrical domains. (English) Zbl 1363.35085

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.


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B51 Comparison principles in context of PDEs
35J25 Boundary value problems for second-order elliptic equations