An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes
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ilan Journal of Mathematics
An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes Francesco Della Pietra and Gianpaolo Piscitelli Abstract. In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem. Mathematics Subject Classification (2010). 35P15, 47J30, 35J92, 35J25. Keywords. Nonlinear eigenvalue problems, torsional rigidity, mixed boundary conditions, optimal estimates.
1. Introduction In this paper we prove an optimal bound for the first eigenvalue and the torsional rigidity of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. The problem of maximizing or minimizing elliptic functionals with different boundary conditions under various geometric constraints has been extensively studied. Based on the classical isoperimetric inequality, the first evidences are the celebrated Faber-Krahn inequality for the first Dirichlet eigenvalue of the Laplacian, or the Saint-Venant principle for the torsional rigidity with Dirichlet boundary conditions. Let us consider Ω and D two bounded open sets, with D convex, Ω Lipschitz, ¯ Here we study the following connected, and D Ω. We will denote by Σ = Ω \ D. problem: p p |ϕ| dσ |∇ϕ| dx + β 1,p Σ ∂D (1.1) , ϕ ∈ W (Σ), ϕ ≡ 0 , λ(β, Σ) = min p |ϕ| dx Σ
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F. Della Pietra Piscitelli F. Della Pietra andand G. G. Piscitelli
where β is a positive constant and 1 < p < +∞. A minimizer u ∈ W 1,p (Σ) of (1.1) satisfies in Σ, −∆p u = λ(β, Σ)|u|p−2 u ∂u |∇u|p−2 =0 on ∂Ω, (1.2) ∂ν |∇u|p−2 ∂u + β|u|p−2 u = 0 on ∂D. ∂ν In this paper we obtain an optimal upper bound on λ(β, Σ), when β > 0 is fixed ¯ varies among all domains such that the volume of Σ and the (n − 1)and Σ = Ω \ D quermassintegral of D are given (see Section 2.3 for the precise definition). If n = 2, the geometrical constraint on D corresponds to fix its perimeter. In particular, we show that λ(β, Σ) is maximized by the spherical shell. The first main result is the following. Theorem 1.1. Let β > 0, and Ω and D be two bounded open sets, with D convex, Ω ¯ A = AR ,R = BR \ B R , where Lipschitz, connected, and D Ω. Let be Σ = Ω \ D, 1 2 2 1 BRi is a ball centered at the origin with radius Ri , i = 1, 2. Suppose that |A| = |Σ|, and Wn−1 (BR1 ) = Wn−1 (D). Then, λ(β, Σ) ≤ λ(β, A). In the case p = n = 2, Theorem 1.1 recovers a result proved by Hersch in [18]. Our result generalizes it to the p-Laplacian operator and to any dimension, giving an answer to a question posed by Henrot in [17, Chapter 3, Open problem 5]. Actually, we are not able to prove or disprove that the first eigenvalue λ(β, Σ) is maximized on the spherical shell when the perimeter of D, and not the Wn−1 , is fixed. We stress that a related result has been recently proved for an optimal insulating problem (see [14]). Optimal esti
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