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

Langford, Jeffrey J.

Differ. Integral Equ. 29, No. 5-6, 493-512 (2016).

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.

Reviewer: Dagmar Medková (Praha)

MSC:

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

Keywords:

Poisson equation; Neumann problem; comparison principle; subharmonicity