On a Conjecture Relative to the Maximum of Harmonic Functions on Convex Domains: Unbounded Domains

Lucio R. Berrone

Abstract: Let $u$ be a harmonic function on a bounded domain $\Omega$ which satisfies the mixed boundary conditions $u|_{\Gamma_{0}}=0$, $\frac{\partial u}{\partial n}|_{\Gamma_{1}}=1$, where $\Gamma_{1}$ is composed by a finite number of subarcs of $\partial\Omega$, $\Gamma_{0}=\partial\Omega\sim\Gamma_{1}$ and $n$ indicates the outward unit normal. In [2] has been conjectured that if $\Omega$ is convex and the subset $\Gamma_{1}$ is made to vary on $\partial\Omega$ while its measure is maintained equal to a constant $C>0$, then $\sup_{x\in\Omega}u$ attains its maximum value when $\Gamma_{1}$ is a certain connected subarc of measure $C$. In the present paper, the case of unbounded domains is discussed.

Keywords: Harmonic functions; mixed boundary value problems; Poisson kernel.

Classification (MSC2000): 35J05, 35B99

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