The Robin eigenvalue problem for the -Laplacian as (original) (raw)

ON THE EIGENVALUE PROBLEM INVOLVING THE ROBIN p(x)-LAPLACIAN

On the Eigenvalue problem involving the Robin p(x)-Laplacian, 2022

In this study, we investigate a nonlinear eigenvalue problem on a bounded domain in R d , d ≥ 2. The existence of a nondecreasing sequence of nonnegative eigenvalues is established using the Ljusternik-Schnirelmann principle. Using the variational principle, we also prove that there exists a principal eigenvalue which is the smallest of all possible eigenvalues and that the set of eigenvalues is not closed.

On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian

Electronic Journal of Differential Equations, 2006

Let Λ p p be the best Sobolev embedding constant of W 1,p (Ω) → L p (∂Ω), where Ω is a smooth bounded domain in R N. We prove that as p → ∞ the sequence Λp converges to a constant independent of the shape and the volume of Ω, namely 1. Moreover, for any sequence of eigenfunctions up (associated with Λp), normalized by up L ∞ (∂Ω) = 1, there is a subsequence converging to a limit function u∞ which satisfies, in the viscosity sense, an ∞-Laplacian equation with a boundary condition.

On the principal eigenvalue of a Robin problem with a large parameter

Mathematische Nachrichten, 2008

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.

Robin Problem Involving the p(x)-Laplacian Operator Without Ambrosetti-Rabinowizt Condition

Robin Problem Involving the p(x)-Laplacian Operator Without Ambrosetti-Rabinowizt Condition, 2024

The paper deals with the following Robin problem      −M Ω 1 p(x) |∇u| p(x) dx + ∂Ω a(x) p(x) |∇u| p(x) dσ div(|∇u| p(x)−2 ∇u) = λh(x, u) in Ω, |∇u| p(x)−2 ∂u ∂ν + a(x)|u| p(x)−2 u = 0 on ∂Ω. The goal is to determine the precise positive interval of λ for which the above problem admits at least two nontrivial solutions without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain Theoreom with Cerami condition.

The Laplacian with Robin boundary conditions on arbitrary domains

2003

Using a capacity approach, we prove in this article that it is always possible to define a realization µ of the Laplacian on L 2 ( ) with generalized Robin boundary conditions where is an arbitrary open subset of R n and µ is a Borel measure on the boundary ∂ of . This operator µ generates a sub-Markovian C 0 -semigroup on L 2 ( ). If dµ = β dσ where β is a strictly positive bounded Borel measurable function defined on the boundary ∂ and σ the (n−1)-dimensional Hausdorff measure on ∂ , we show that the semigroup generated by the Laplacian with Robin boundary conditions β has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L p ( ) is independent of p ∈ [1, ∞). Our approach constitutes an alternative way to Daners who considers the (n − 1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture. : 31C15, 35J25, 46E35, 47D03, 47D07.

Spectral stability of the Robin Laplacian

Proceedings of the Steklov Institute of Mathematics, 2008

We consider the Robin Laplacian in two bounded regions Ω 1 and Ω 2 of R N with Lipschitz boundaries and such that Ω 2 ⊂ Ω 1 , and we obtain two-sided estimates for the eigenvalues λ n,2 of the Robin Laplacian in Ω 2 via the eigenvalues λ n,1 of the Robin Laplacian in Ω 1. Our estimates depend on the measure of the set difference Ω 1 \ Ω 2 and on suitably defined characteristics of vicinity of the boundaries ∂Ω 1 and ∂Ω 2 , and of the functions defined on ∂Ω 1 and on ∂Ω 2 that enter the Robin boundary conditions.

Eigenvalues of −(∆ p + ∆ q ) under a Robin-like boundary condition *

2016

Let Ω ⊂ IR N , N ≥ 2, be a bounded open set with smooth boundary. Consider in Ω the equation −∆ p u − ∆ q u = λ|u| p−2 u subject to a Robin-like boundary condition involving a positive constant α, where p, q ∈ (1, ∞), p = q, and λ ∈ IR. We show that there is no eigenvalue λ of the above problem in the interval (−∞, λ R ], where λ R := inf { Ω ||v| p dx+α ∂Ω |v| p ds; v ∈ W 1,max{p,q} (Ω), Ω |v| p dx = 1}, while any λ ∈ (λ R , λ *) is an eigenvalue of this problem, where λ * := αm N −1 (∂Ω)/m N (Ω). Note that the case p = q investigated here is complementary to the homogeneous case p = q for which the set of eigenvalues is completely known only if p = q = 2. MSC: 35J60, 35J92, 49R05 keywords: Robin-like eigenvalue problem, p-Laplacian, Sobolev space, Nehari type manifold, variational methods.

The Robin–Laplacian problem on varying domains

Calculus of Variations and Partial Differential Equations, 2016

We prove a stability result for elliptic equations under general Dirichlet-Robin boundary conditions with respect to the variation of the domain under the Hausdorff complementary topology. As a by-product, under the additional assumption of the convergence of the perimeters, we obtain a stability result for the classical Robin-Laplacian. Contents 1. Introduction 1 2. Notation and Preliminaries 4 3. The main results 6 4. Proof of the stability results 9 4.1. Some technical lemmas 9 4.2. The stability results for the general problem 12 4.3. Proof of the stability result for the classical Robin problems 15 5. Proof of the Γ-convergence result 16 References 24