PORTUGALIAE MATHEMATICA Vol. 55, No. 3, pp. 307321 (1998) 

On a Conjecture Relative to the Maximum of Harmonic Functions on Convex Domains: Unbounded DomainsLucio R. BerroneAbstract: 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 Full text of the article:
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