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- W2092242805 abstract "We consider a harmonic function u defined on a bounded domain $Omega $ $% subset Bbb{R}^{2}$ and satisfying the mixed boundary conditions $% u|_{Gamma _{0}}=0, (partial u/partial n)|_{Gamma _{1}}=1$, where $% Gamma _{1}$ is composed by a finite number of arcs of $partial Omega $ and $Gamma _{0}=partial Omega sim overline{Gamma _{1}}$. In [Berrone, Subsistencia de Modelos Matematicos que Involucran a la Ecuacion del Calor-Difusion, Ph.D. thesis, Universidad Nacional de Rosario, Argentina, 1994] it was conjectured that if $Omega $ is convex and the subset $% Gamma _{1}$ is made to vary on $partial Omega $ so as to maintain its measure equal to a constant C>0, then $Gamma _{1}mapsto supnolimits_{xin Omega }u$ attains its maximum value when $Gamma _{1}$ is a certain connected arc of measure C. The present paper has evolved from attempts to prove this conjecture. When certain geometric restrictions are satisfied by the components of $Gamma _{1}$, the property stated by the conjecture is shown to hold for every regular domain $Omega $, convex or not, and every connected arc, provided that the measure $% left| Gamma _{1}right| $ is sufficiently small (see Theorem ref{referee}). However, convexity becomes a necessary condition in order that the full conjecture can be supportable (see section 2). In addition, some variations of the conjecture are proposed." @default.
- W2092242805 created "2016-06-24" @default.
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- W2092242805 date "1999-01-01" @default.
- W2092242805 modified "2023-09-26" @default.
- W2092242805 title "On a Conjecture Relative to the Maxima of Harmonic Functions on Convex Domains" @default.
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- W2092242805 doi "https://doi.org/10.1137/s0036141098334973" @default.
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