Sharp Bounds for the p-Torsion of Convex Planar Domains
We obtain some sharp estimates for the p-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on t
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Abstract We obtain some sharp estimates for the p-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behavior of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimization problem which consists in maximizing the p-torsion among polygons having a given number of vertices and a given area. A long-standing conjecture by PólyaSzegö states that the solution is the regular polygon. We show that such conjecture is true within the subclass of polygons for which a suitable notion of “asymmetry measure” exceeds a critical threshold. Keywords Isoperimetric inequalities · Shape optimization · Web functions · Convex shapes
1 Introduction Let Ω ⊂ R2 be an open bounded domain and let p ∈ (1, +∞). Consider the boundary value problem −Δp u = 1 in Ω (1) u=0 on ∂Ω,
I. Fragalà (B) · F. Gazzola Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy e-mail: [email protected] F. Gazzola e-mail: [email protected] J. Lamboley Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_7, © Springer-Verlag Italia 2013
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where Δp u = div(|∇u|p−2 ∇u) denotes the p-Laplacian. The p-torsion of Ω is defined by p τp (Ω) := |∇up | = up , (2) Ω
Ω 1,p W0 (Ω).
being up the unique solution to (1) in Notice that the second equality in (2) is obtained by testing (1) by up and integrating by parts. Since (1) is the Euler-Lagrange equation of the variational problem 1 p |∇u| − u , min Jp (u), where Jp (u) = (3) 1,p Ω p u∈W (Ω) 0
there holds τp (Ω) =
p min Jp (u). 1 − p u∈W 1,p (Ω) 0
A further characterization of the p-torsion is provided by the equality τp (Ω) = p S(Ω)1/(p−1) , where S(Ω) is the best constant for the Sobolev inequality uL1 (Ω) ≤ p
1,p
S(Ω)∇uLp (Ω) on W0 (Ω). The purpose of this paper is to provide some sharp bounds for τp (Ω), holding for a convex planar domain Ω, in terms of its area, perimeter, and inradius (in the sequel denoted respectively by |Ω|, |∂Ω|, and RΩ ). The original motivation for studying this kind of shape optimization problem draws its origins in the following long-standing conjecture by Pólya and Szegö: Among polygons with a given area and N vertices, the regular N -gon maximizes τp .
(4)
A similar conjecture is stated by the same Authors also for the principal frequency and for the logarithmic capacity, see [15]. For N = 3 and N = 4 these conjectures were proved by Pólya and Szegö themselves [15, p. 158]. For N ≥ 5, to the best of our knowledge, the unique solved case is the one of logarithmic capacity, see the beautiful paper [16] by Solynin and Zalgaller; the cases of torsion and principa
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