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The inradius, the first eigenvalue, and the torsional rigidity of curvilinear polygons
Author(s) -
Solynin Alexander Yu.,
Zalgaller Victor A.
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq028
Subject(s) - mathematics , curvilinear coordinates , incircle and excircles of a triangle , isoperimetric inequality , mathematical proof , rigidity (electromagnetism) , mathematical analysis , laplace operator , polygon (computer graphics) , eigenvalues and eigenvectors , curvature , combinatorics , pure mathematics , geometry , physics , structural engineering , quantum mechanics , engineering , telecommunications , frame (networking) , computer science
Let λ 1 , P , and ρ denote the first eigenvalue of the Dirichlet Laplacian, the torsional rigidity, and the inradius of a planar domain Ω, respectively. In this paper, we prove several inequalities for λ 1 , P , and ρ in the case when Ω is a curvilinear polygon with n sides, each of which is a smooth arc of curvature at most κ . Our main proofs rely on the method of dissymmetrization and on a special geometrical ‘containment theorem’ for curvilinear polygons. For rectilinear n ‐gons, which constitute a proper subclass of curvilinear n ‐gons with curvature at most 0, these inequalities were established by the first author in 1992. In the simplest particular case of Euclidean triangles T , the inequality linking the first eigenvalue and the inradius of T is equivalent to the inequality λ 1 ⩽ π 2 L 2 /9 A 2 , where A and L denote the area and perimeter of T , respectively, which was recently discussed by Freitas ( Proc. Amer. Math. Soc. 134 (2006) 2083–2089) and Siudeja ( Michigan Math. J. 55 (2007) 243–254).