An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes

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.

A monotonicity result for the first Steklov–Dirichlet Laplacian eigenvalue

“…Specifically, different boundary conditions can be imposed on the outer and inner boundary and hence several optimization problems can be studied (e.g. Robin-Neumann [18], Neumann-Robin [7], Dirichlet-Neumann [2,3], Steklov-Dirichlet [11,14,17], Steklov-Robin [12]). At the mean time, the optimal placement of an obstacle has been studied, so as to maximize or minimize a prescribed functional (e.g.…”

Piecewise nonlinear materials and Monotonicity Principle

“…Optimization problems on domains with holes have received considerable interest in recent years, see e.g. [22,27,47,48] and reference therein.…”

Reverse Faber-Krahn inequalities for Zaremba problems

“…In [1], for n ≥ 3 and Ω as given in (4), Anoop and Ashok observed that the inequality (11) holds for Γ = Γ D when Ω D is a ball in R n . As a consequence, they obtained (9) for the first eigenvalue τ 1,q (Ω) of (P) with q = p in higher dimensions under the assumption that Ω D is a ball; cf. [1, Theorem 1.2].…”

“…In [9], Della Pietra and Piscitelli established the inequality part in Theorem 1.6 (see [9, Theorem 1.1, Remark 3.1]) using a web function and the co-area formula. Here we use the higher dimensional analogue (see Corollary 3.4) of Sz.…”