Eigenvalue estimates for submanifolds with bounded <Emphasis Type="BoldItalic">f</Emphasis>-mean curvature

In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the f-Laplacian on complete noncompact submanifolds of the weighted Riemannian manifold (H ^m (?1),e^?f dv) with respect to the f-mean curvature. In particular, our results generalize those of Cheung and Leung in Math. Z. 236 (2001) 525–530.     

Existence of nonnegative viscosity solutions for a class of problems involving the $${\infty}$$-Laplacian

The issue of existence of nonnegative solutions for a class of problems depending on a real parameter and involving the ∞-Laplacian is considered. The problem is treated as the limiting case of a family of perturbed eigenvalue problems for the (p, q(p))-Laplacian as \({p\rightarrow\infty}\). It is shown that nontrivial nonnegative viscosity solutions for this class of problems exist if and only if the parameter is greater than or equal to the reciprocal of the maximum of the distance to the boundary of the domain.     

A new characterization of Willmore submanifolds

Let M ⁿ be a compact Willmore submanifold in the unit sphere S ^n+p . In this note, we investigate the first eigenvalue of the Schrödinger operator L = ?Δ?q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.     

The <Emphasis Type="Italic">p</Emphasis>-Harmonic Measure of Small Axially Symmetric Sets

Using the first eigenvalue/eigenvector pair of a singular eigenvalue problem (motivated by the Dirichlet eigenvalue problem for the Laplace-Beltrami operator on a spherical cap), we define certain nonnegative p-superharmonic and p-subharmonic functions on a convex cone which are singular at the vertex and vanish on the rest of the boundary. We use these functions to give upper and lower estimates of the p-harmonic measure near the vertex of the cone as well as the p-harmonic measure of a small spherical cap.     

Hardy Inequalities for Finsler <Emphasis Type="Italic">p</Emphasis>-Laplacian in the Exterior Domain

The Finsler p-Laplacian is the class of nonlinear differential operators given by

$$\begin{aligned} \Delta _{H,p}u:= \text {div}(H(\nabla u)^{p-1}\nabla _{\eta } H(\nabla u)) \end{aligned}$$

where \(1<p<\infty \) and \(H:\mathbf {R}^N\rightarrow [0,\infty )\) is in \(C^2(\mathbf {R}^N\backslash \{0\})\) and is positively homogeneous of degree 1. Under some additional constraints on H, we derive the Hardy inequality for Finsler p-Laplacian in exterior domain for \(1<p\le N\). We also provide an improved version of Hardy inequality for the case \(p=2\).

     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.

     

Existence of saddle solutions of a nonlinear elliptic equation involving <Emphasis Type="Italic">p</Emphasis>-Laplacian in more even-dimensional spaces

We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p > 2, -Δ _p u = f(u) in R^2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m ≥ 2p.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

Global continuum of positive solutions for discrete <Emphasis Type="Italic">p</Emphasis>-Laplacian eigenvalue problems

We discuss the discrete p-Laplacian eigenvalue problem,

$$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$$

where T > 1 is a given positive integer and φ _p (x):= |x| ^p?2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.

     

Hadamard variational formulas for <Emphasis Type="Italic">p</Emphasis>-torsion and <Emphasis Type="Italic">p</Emphasis>-eigenvalue with applications

In this paper, we pose two kinds of Minkowski problems involving the p-Laplacian operator. The Hadamard variational formulas for some p-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of p-Laplacian equations.     

On the continuity of solutions to elliptic equations with variable order of nonlinearity

We study the p-Laplacian with variable exponent p(x) bounded away from unity and infinity. We obtain a sufficient condition on p(x) under which all solutions of the p-Laplace equation are continuous at a fixed point of a domain, and find an estimate for the modulus of continuity of solutions.     

Compactness of the Space of Minimal Hypersurfaces with Bounded Volume and <Emphasis Type="Italic">p</Emphasis>-th Jacobi Eigenvalue

Given a closed Riemannian manifold of dimension less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for \(p\ge 2\) one gains strong convergence to a smooth limit submanifold away from at most \(p-1\) points.     

The existence of semiclassical states for some <Emphasis Type="Italic">p</Emphasis>-Laplacian equation with critical exponent

In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε ≤ E; for any m ∈ ?, it has m pairs solutions if ε ≤ E _m , where E, E_m are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W^1,p(? ^N ) as ε → 0.     

Interior regularity of space derivatives to an evolutionary,symmetric $$\varphi $$-Laplacian

We consider the Orlicz-growth generalization to the evolutionary p-Laplacian and to the evolutionary symmetric p-Laplacian. We derive the spatial second-order Caccioppoli-type estimate for a local weak solution to these systems. Our result is new even for the p-case.     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \(\left[ {\frac{{n + 1}}{2}} \right]\) or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S³, F) if any prime closed geodesic has non-zero Morse index.     

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume.The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M ⁿ , how “fast” can this sphere be contracted to a point, if π _i (M ⁿ )={0} for 1≤i<k. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible k-dimensional sphere in a (k-1)-connected manifold M ⁿ providing that M ⁿ is not k-connected?     

On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n ≥ 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically.     

Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices

Let X ⁽ⁿ⁾=(X _ij ) be a p×n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R ⁽ⁿ⁾=(ρ _ij ) be the p×p sample correlation coefficient matrix of X ⁽ⁿ⁾, and \(S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}\) be the sample covariance matrix of X ⁽ⁿ⁾, where \(\bar{X}\) is the mean vector of the n observations. Assuming that X _ij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R ⁽ⁿ⁾ converges almost surely to the limit \((1-\sqrt{c}\,)^{2}\) as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S ⁽ⁿ⁾ converges almost surely to \((1-\sqrt{c}\,)^{2}\).     

On a Harnack inequality for the elliptic (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian

A special Harnack inequality is proved for solutions of nonlinear elliptic equations of the p(x)-Laplacian type with a variable exponent p(x) that takes different values on two sides of a hyperplane dividing the domain. Examples are given showing that the classical Harnack inequality does not hold in this case.