The inradius, the first eigenvalue, and the torsional rigidity of curvilinear polygons

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).