Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Upper and lower bounds for the first Robin eigenvalue of nonlinear elliptic operators

Rosa Barbato, Francesco Della Pietra

TL;DR

The paper addresses the problem of bounding the first Robin eigenvalue $\lambda_p(\beta,\Omega)$ of the nonlinear $p$-Laplacian under Robin boundary conditions. It develops a Thompson-type variational framework and analyzes asymptotics as $\beta$ varies, yielding explicit upper bounds in terms of the Dirichlet eigenvalue and the $p$-torsional rigidity, as well as a general lower bound via a convexity argument. The results extend classical bounds for the Laplacian (e.g., Sperb, Pólya) to the nonlinear setting and connect spectral data to geometric quantities like $P(\Omega)$ and $T_p(\Omega)$. Overall, the work provides practical estimates for $\lambda_p(\beta,\Omega)$ and establishes structural links between Robin, Dirichlet, and Neumann-type problems through variational principles and limit analyses.

Abstract

Let $Ω$ be a bounded, smooth domain of $\mathbb R^N$, $N\ge 2$. In this paper, we prove some inequalities involving the first Robin eigenvalue of the $p$-laplacian operator. In particular, we prove an upper bound for the first Robin eigenvalue of nonlinear elliptic operators in terms of the first Dirichlet eigenvalue.

Upper and lower bounds for the first Robin eigenvalue of nonlinear elliptic operators

TL;DR

The paper addresses the problem of bounding the first Robin eigenvalue of the nonlinear -Laplacian under Robin boundary conditions. It develops a Thompson-type variational framework and analyzes asymptotics as varies, yielding explicit upper bounds in terms of the Dirichlet eigenvalue and the -torsional rigidity, as well as a general lower bound via a convexity argument. The results extend classical bounds for the Laplacian (e.g., Sperb, Pólya) to the nonlinear setting and connect spectral data to geometric quantities like and . Overall, the work provides practical estimates for and establishes structural links between Robin, Dirichlet, and Neumann-type problems through variational principles and limit analyses.

Abstract

Let be a bounded, smooth domain of , . In this paper, we prove some inequalities involving the first Robin eigenvalue of the -laplacian operator. In particular, we prove an upper bound for the first Robin eigenvalue of nonlinear elliptic operators in terms of the first Dirichlet eigenvalue.

Paper Structure

This paper contains 6 sections, 18 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. A characterization for TEXT
  4. Limit properties when TEXT or TEXT
  5. Upper bounds for TEXT
  6. A lower bound for TEXT

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^N$ be an open bounded set, with $C^{1,\gamma}$ boundary, and let $\beta$ be a positive number. Then, where $\bar{K}$ is given by and $v$ is a first eigenfunction of $\lambda^{D}_{p}(B)$, defined in a ball $B$ such that $\lambda^{D}_{p}(B)=\lambda^{D}_{p}(\Omega)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 25 more