First eigenvalue of the <Emphasis Type="Italic">p</Emphasis>-Laplace operator along the Ricci flow

In this article, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the p-Laplace operator along the Ricci flow on closed manifolds. We show that the first p-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first p-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a p-eigenvalue comparison-type theorem when its Euler characteristic is negative.     

Evolution of the first eigenvalue of the clamped plate on manifold along the Ricci flow

ABSTRACT

In this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds.     

The First Eigenvalue of p-Laplace Operator Under Powers of the mth Mean Curvature Flow

In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let ( ${M^{n}, g(t)), t\in [0,T),}$ be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ_1,p (t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and $$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$ then λ_1,p (t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow.     

An alternative approach to the Faber–Krahn inequality for Robin problems

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions. The techniques introduced allow to work with much less regular domains by using test function arguments. We substantially simplify earlier proofs, and establish the sharpness of the inequality for a larger class of domains at the same time.     

Faber-Krahn inequality for robin problems involving <Emphasis Type="Italic">p</Emphasis>-Laplacian

The eigenvalue problem for the p-Laplace operator with Robin boundary conditions is considered in this paper.A Faber-Krahn type inequality is proved.More precisely,it is shown that amongst all the domains of fixed volume,the ball has the smallest first eigenvalue.     

Convergence of Inverse Power Method for First Eigenvalue of p-Laplace Operator

In this article, convergence of an iterative scheme to approximate the first eigenfunction and related eigenvalue for p-Laplace operator is shown. Moreover, numerical examples are presented that show the e?ciency and accuracy of the algorithm.     

Rigidity of manifolds with Bakry–émery Ricci curvature bounded below

Let M be a complete Riemannian manifold with Riemannian volume vol _g and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator ${\Delta_f = \Delta- \nabla f \cdot \nabla}$ was given by Wu (J Math Anal Appl 361:10?C18, 2010) and Wang (Ann Glob Anal Geom 37:393?C402, 2010) under a condition that finite dimensional Bakry?Cémery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li?CWang??s result in Wang (J Differ Geom 58:501?C534, 2001, J Differ Geom 62:143?C162, 2002). For the case that infinite dimensional Bakry?CEmery Ricci curvature of M is bounded below, we do not expect any upper bound estimate on the first eigenvalue of ?? _f without any additional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of ?? _f under the additional assuption that ${|\nabla f|}$ is bounded. We also obtain the rigidity result on this estimate, as another Li?CWang type splitting theorem.     

Eigenvalue problems for the p-Laplacian

We study nonlinear eigenvalue problems for the p$p$-Laplace operator subject to different kinds of boundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.     

Gradient estimates on the weighted p-Laplace heat equation

In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces

New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry–Émery Ricci curvature, the Escobar–Lichnerowicz–Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry–Émery Ricci curvature and the m-Bakry–Émery Ricci curvature bounded from below by a non-positive constant, the Li–Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results.     

典型几何上Ricci流下的特征值

研究了典型几何上规范Ricci流下Laplace-Beltrami算子第一特征值的发展行为.在每一个Bianchi类中,我们估计了特征值的导数.构造了Ricci流下的单调量并得到了特征值的上下界估计.     

The first eigenfunctions and eigenvalue of the p-Laplacian on Finsler manifolds

This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C~(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.     

Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds

In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace–Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.     

On local Morse theory for p-area functionals,p > 2

This survey article deals with some Morse theoretic aspects for functionals defined in Sobolev Banach spaces, associated with quasilinear elliptic equations or systems, involving the p-Laplace operator, p > 2.We discuss the notion of nondegeneracy in a Banach (not Hilbert) variational framework and we present some developments concerning the critical groups estimates and the interpretation of the multiplicity of a critical point.     

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p  -Laplacian (1<p<∞$1<p<\infty$) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].     

Degenerate elliptic equations with singular nonlinearities

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of , and compactness holds below a critical dimension N ^#. The nonlinearity f(u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p ≠ 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blow-up argument and stronger assumptions on the nonlinearity f(u) are required. Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”.     

Isoperimetric inequalities,Wulff shape and related questions for strongly nonlinear elliptic operators

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p--Laplace operator $\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert\nabla u \vert^{p-2}{{\partial u}\over{\partial x_i}})$, the pseudo-p-Laplace operator $\tilde\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert {{\partial u}\over{\partial x_i}} \vert^{p-2} {{\partial u}\over{\partial x_i}})$ and others. We derive the positivity of the first eingefunction, simlicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H ^o. They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

A priori estimates for the difference of solutions to quasi-linear elliptic equations

The purpose of this paper is to prove a priori estimates for the difference of weak solutions to certain quasi-linear elliptic equations, where the p(x)-Laplace operator serves as a prototype for our method.