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

Let p 1 and p 0 be closed, regular, convex, concentric polygons having n sides in ℝ 2 such that the circumradius of p 0 is strictly less than the inradius of p 1 . We fix p 1 and vary p 0 by rotating it about its center. Let Ω be the interior of p 1 p 0 . Let u be the solution of the stationary prob...

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Bibliographic Details
Published in:Indian journal of pure and applied mathematics 2014-12, Vol.45 (6), p.861-874
Main Authors: Aithal, A. R., Sarswat, Acushla
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematics
Mathematics and Statistics
Numerical Analysis
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Summary:Let p 1 and p 0 be closed, regular, convex, concentric polygons having n sides in ℝ 2 such that the circumradius of p 0 is strictly less than the inradius of p 1 . We fix p 1 and vary p 0 by rotating it about its center. Let Ω be the interior of p 1 p 0 . Let u be the solution of the stationary problem −Δ u = 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 p 0 coincide with those of p 1 .
ISSN:0019-5588
0975-7465
DOI:10.1007/s13226-014-0094-3