Eigenfunctions for singular fully nonlinear equations in unbounded domains

In this paper, we prove the existence of a generalized eigenvalue and a corresponding eigenfunction for fully nonlinear elliptic operators singular or degenerate, homogeneous of degree 1+α, α > −1 in unbounded domains of IR ^N . The main tool will be Harnack’s inequality.     

On the eigenvalue counting function for weighted Laplace-Beltrami operators

A weighted Laplace-Beltrami operator on a Riemannian manifold M is an operator of the form

Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ?r-1, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2r. We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.     

An algorithm for numerical determination of the structure of a general matrix

An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.An adaptation of the algorithm is developed which, applied to a matrix with a very ill-conditioned eigenproblem, computes a degenerate matrix. The difference between these matrices is small, measured in the spectral norm. The degenerate matrix will appear in a standard form, whose eigenvalues and principal vectors can be computed in a numerically stable manner.Numerical examples are given.     

Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2

The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of on an open bounded subset with smooth boundary as B tends to infinity. We introduce a “magnetic” curvature mixing the curvature of ∂Ω and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg–Landau theory in the variable magnetic field case. Submitted: June 26, 2008., Accepted: November 28, 2008.     

Persistence of embedded eigenvalues

We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m<∞ we show that in favorable situations, the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of codimension m. We also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite.     

On the principal eigenvalue of degenerate quasilinear elliptic systems

We study the properties of the positive principal eigenvalue of a degenerate quasilinear elliptic system. We prove that this eigenvalue is simple, unique up to positive eigenfunctions and isolated. Under certain restrictions on the given data, the regularity of the corresponding eigenfunctions is established. The extension of the main result in the case of an unbounded domain is also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The eigenvalue distribution for a degenerate elliptic operator — div {(1−|x|2)grad·}

We sharpen the remainder estimate of the asymptotic formula for the eigenvalue distribution for the degenerate operator -div{(1−|x|²)grad·} in the unit ball of the Euclidean spaceR ⁿ . In particular we find the second term whenn=2.     

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

In this paper, we first prove the CR analogue of Obata’s theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau’s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P ₀. Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator. This research was supported in part by the NSC of Taiwan.     

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.      

Higher Integrability of the Gradient in Degenerate Elliptic Equations

We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta–Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

On positivity of solutions of degenerate boundary value problems for second-order elliptic equations

In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B _u =gon Ω∂Г where ω is a domain in ℝ ⁿ ,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary. The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ=∂Ω, were examined intensively by many authors.     

Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived.The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than π/3. Antisymmetry is proved for apertures greater than π/3.     

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μ _t be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between μ_t and μ is not small for t ≪λ ⁻¹ logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.     

一类三阶奇摄动特征值问题

将带有边界条件的三阶非齐次线性方程的可解性条件应用到退化特征值问题上,得出了奇摄动问题的解的渐近表示式.     

On the absence of the basis property for the root function system of the Sturm–Liouville operator with degenerate boundary conditions

The eigenvalue problem generated by the Sturm–Liouville equation on the interval (0, π) with degenerate boundary conditions is considered. Under certain conditions imposed on the spectrum, it is shown that the system of eigen- and associated functions is not a basis in L₂(0, π).     

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.     

A weighted eigenvalue problem for the p-Laplacian plus a potential

Let Δ _p denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem

Decay law for a quasistationary state of the Schrödinger operator for a crystal film

For the Schrödinger operator in a cell corresponding to a crystal film pattern, eigenvalues may exist in the continuous spectrum and become resonances under perturbations. We prove that the corresponding decay law in a nonstationary approach is exponential for a nondegenerate (in some cases, degenerate) eigenvalue.