Boundary slopes and the logarithmic limit set

The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected boundary curves of a multi-cusped manifold in a similar way to the 1-cusped case, where the slopes are encoded in the Newton polygon of the A-polynomial.     

Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities

Under study are the two classes of elliptic spectral problems with homogeneous Dirichlet conditions and discontinuous nonlinearities (the parameter occurs in the nonlinearity multiplicatively). In the former case the nonlinearity is nonnegative and vanishes for the values of the phase variable not exceeding some positive number c; it has linear growth at infinity in the phase variable u and the only discontinuity at u = c. We prove that for every spectral parameter greater than the minimal eigenvalue of the differential part of the equation with the homogeneous Dirichlet condition, the corresponding boundary value problem has a nontrivial strong solution. The corresponding free boundary in this case is of zero measure. A lower estimate for the spectral parameter is established as well. In the latter case the differential part of the equation is formally selfadjoint and the nonlinearity has sublinear growth at infinity. Some upper estimate for the spectral parameter is given in this case.     

Bifurcation from Infinity for Asymptotically Linear Elliptic Eigenvalue Problems

In this paper we are going to discuss bifurcation from infinity for asymptotically linear elliptic eigenvalue problems having nonlinear boundary conditions. Behavior of the bifurcation components is also studied.     

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.     

Coincidence degree and nontrivial solutions of elliptic boundary value problems at resonance

Two results on the computation of the coincidence degree at infinity or at θ are presented. Then they are applied to the investigation of nontrivial solutions for elliptic boundary value problems in a case where the problems are resonant both at θ and at infinity.     

Darboux Transformations Associated with Boiti-Tu Eigenvalue Problem

To the hierarchy of nonlinear evolution equations which associate with Boiti-Tu eigenvalue problem, an explicit and universal form of Baeklund transformations and auniversal proof are presented. It is called Darboux transformation. By this method, to ask for a new solution of every system of equations of the hierarchy, it is sufficient to solve some linear problems. Here the constraints at the boundary for the potentials (for example, at x=\pm \infinity) are removed.     

Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. PerronFrobenius theory ensures the existence of a corresponding positive eigenvector ψ. The goal of the paper is to give bounds on the amplitude max ψ/ min ψ. Two approaches are proposed: One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo(2014).     

Layer potential techniques for the narrow escape problem

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.     

Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate

We consider a prototype reaction-diffusion system which models a network of two consecutive reactions in which chemical components A and B form an intermediate C which decays into two products P and Q. Such a situation often occurs in applications and in the typical case when the intermediate is highly reactive, the species C is eliminated from the system by means of a quasi-steady-state approximation. In this paper, we prove the convergence of the solutions in L², as the decay rate of the intermediate tends to infinity, for all bounded initial data, even in the case of initial boundary layers. The limiting system is indeed the one which results from formal application of the QSSA. The proof combines the recent L²-approach to reaction-diffusion systems having at most quadratic reaction terms, with local L^∞-bounds which are independent of the decay rate of the intermediate. We also prove existence of global classical solutions to the initial system.     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

Natural solutions of rational Stieltjes moment problems

In the strong or two-point Stieltjes moment problem, one has to find a positive measure on [0,∞) for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions μ₀ and μ_∞ exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points.     

Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1≤j≤n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.     

Dynamics of Quasi-parabolic One-Resonant Biholomorphisms

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of ? ⁿ⁺¹ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one-dimensional family of resonant relations. We define some invariants and give conditions which ensure the existence of attracting domains for such maps.     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.     

Nonlinear Riemann - Hilbert Problems without Transversality

Nonlinear Riemann - Hilbert problems (RHP) generalize two fundamental classical problems for complex analytic functions, namely: 1. the conformal mapping problem, and 2. the linear Riemann - Hilbert problem. This paper presents new results on global existence for the nonlinear (RHP) in doubly connected domains with nonclosed restriction curves for the boundary data. More precisely, our nonlinear (RHP) is required to become ?at infinity”?, i.e., for solutions having large moduli, a linear (RHP) with variable coefficients. Global existence for q-connected domains was already obtained in [9] for the special case that the restriction curves for the boundary data ?at infinity”? coincide with straight lines corresponding to linear (RHP)-s with special so-called constant - coefficient transversality boundary conditions. In this paper, the boundary conditions are much more general including highly nonlinear conditions for bounded solutions in the context of nontransversality. In order to prove global existence, we reduce the problem to nonlinear singular integral equations which can be treated by a degree theory of Fredholm - quasiruled mappings specifically constructed for mappings defined by nonlinar pseudodifferential operators.     

Gradient estimates and the first Neumann eigenvalue on manifolds with boundary

By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on non-convex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via dimension, radius and bounds of the curvature and the second fundamental form. Finally, some new estimates are also presented for the strictly convex case.     

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.     

Stationary solitons of a three-wave model generated by Type II second-harmonic generation in quadratic media

In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established.