On a functional connected to the laplacian in a family of punctured regular polygons in ℝ2

A. R. Aithal () and Acushla Sarswat ()
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A. R. Aithal: University of Mumbai
Acushla Sarswat: University of Mumbai

Indian Journal of Pure and Applied Mathematics, 2014, vol. 45, issue 6, 861-874

Abstract: Abstract Let p1 and p0 be closed, regular, convex, concentric polygons having n sides in ℝ2 such that the circumradius of p0 is strictly less than the inradius of p1. We fix p1 and vary p0 by rotating it about its center. Let Ω be the interior of p1 p0. Let u be the solution of the stationary problem −Δu = 1 in Ω vanishing on the boundary. We show that the associated Dirichlet energy functional J(Ω) attains its extremum values when the axes of symmetry of p0 coincide with those of p1.

Keywords: Laplacian operator; boundary value problems for second order elliptic equations; variational methods for second order elliptic equations; second order elliptic equations; variational methods (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s13226-014-0094-3

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