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ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.

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Title: ON A CONJECTURE RELATIVE TO THE MAXIMA OF HARMONIC FUNCTIONS ON CONVEX DOMAINS.
Authors: Berrone, Lucio R.1 berrone@unrctu.edu.ar
Source: SIAM Journal on Mathematical Analysis. 1999, Vol. 30 Issue 6, p1185-1207. 23p. 5 Diagrams.
Subjects: Harmonic functions, Fourier series, Harmonic analysis (Mathematics), Boundary value problems, Convex surfaces
Abstract: We consider a harmonic function u defined on a bounded domain Ω ⊂ R² and satisfying the mixed boundary conditions u|Γ0 = 0; (∂u/∂n)|Γ1 = 1, where Γ1 is composed by a finite number of arcs of ∂Ω and Γ0 = ∂Ω ∼ Γ1. 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 Ω is convex and the subset Γ1 is made to vary on ∂Ω so as to maintain its measure equal to a constant C > 0, then Γ1 ... u attains its maximum value when Γ1 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 Γ1, the property stated by the conjecture is shown to hold for every regular domain Ω, convex or not, and every connected arc, provided that the measure |Γ1| is sufficiently small (see Theorem 5). 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. [ABSTRACT FROM AUTHOR]
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Abstract:We consider a harmonic function u defined on a bounded domain Ω ⊂ R² and satisfying the mixed boundary conditions u|Γ0 = 0; (∂u/∂n)|Γ1 = 1, where Γ1 is composed by a finite number of arcs of ∂Ω and Γ0 = ∂Ω ∼ Γ1. 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 Ω is convex and the subset Γ1 is made to vary on ∂Ω so as to maintain its measure equal to a constant C > 0, then Γ1 ... u attains its maximum value when Γ1 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 Γ1, the property stated by the conjecture is shown to hold for every regular domain Ω, convex or not, and every connected arc, provided that the measure |Γ1| is sufficiently small (see Theorem 5). 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. [ABSTRACT FROM AUTHOR]
ISSN:00361410
DOI:10.1137/S0036141098334973