In the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in Alvino et al. (Commun Pure Appl Math, to appear) to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in Alvino et al. (to appear) only in the planar case, holds true in any dimension if p is sufficiently small.
Let $$\Omega \subset \mathbb {R}^2$$ Ω ⊂ R 2 be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to $$\Omega $$ Ω with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if $$\Omega $$ Ω is a disk and the torsion function u is radial.
We study the behaviour, when p → +∞, of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞-Laplacian.Moreover, in the second part of the paper, we focus our attention on the p-Poisson equation when the datum f belongs to L ∞ (Ω) and we study the behaviour of solutions when p → ∞.
In this paper, we prove an upper bound for the first Robin eigenvalue of the p-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the p-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter.The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger [PW61] and Polya [P 60].MSC 2020: 46E30, 35A23, 35J92.
We study the behaviour, when p → + ∞ p\to +\infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{\infty }\left(\Omega ) and we study the behaviour of solutions when p → ∞ p\to \infty .
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