Positive Eigenfunction Research Articles

Existence of solutions for a parabolic equation in bounded domain

In this paper, we study the existence of nonnegative solutions for the parabolic problemwhere D is a smooth domain of , (), the function is such that and is the normalized positive eigenfunction corresponding to the first eigenvalue of and V belongs to a class of time-dependant potentials which can be written as a nonlinear combination of derivatives of a function.

On eigenvalue problems arising from nonlocal diffusion models

We aim at saying as much as possible about the spectra of three classes of linear diffusion operators involving nonlocal terms. In all but one cases, we characterize the minimum \begin{document} $λ_p$ \end{document} of the real part of the spectrum in two max-min fashions, and prove that in most cases \begin{document} $λ_p$ \end{document} is an eigenvalue with a corresponding positive eigenfunction, and is algebraically simple and isolated; we also prove that the maximum principle holds if and only if \begin{document} $λ_p>0$ \end{document} (in most cases) or \begin{document} $≥ 0$ \end{document} (in one case). We prove these results by an elementary method based on the strong maximum principle, rather than resorting to Krein-Rutman theory as did in the previous papers. In one case when it is impossible to characterize \begin{document} $λ_p$ \end{document} in the max-min fashion, we supply a complete description of the whole spectrum.

Existence of solutions for electron balance problem in the stationary radio-frequency induction discharges

A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of an eigenvalue problem with nonlinear dependence on the parameter for a second order ordinary differential equation is established. The initial problem is approximated by the finite element method. Error estimates for the approximate minimal eigenvalue and corresponding positive eigenfunction are derived. Problems of this form arise in modelling the plasma of a radio-frequency discharge at reduced pressure.

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for the existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Martin compactification of a complete surface with negative curvature

In this paper we consider the Martin compactification, associated with the operator $$\mathcal {L}= \Delta -1$$ , of a complete non-compact surface $$\Sigma ^2$$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $$\Delta $$ of $$\Sigma ^2$$ , and prove a uniqueness result: such eigenfunctions are unique up to a positive multiplicative constant, if they vanish on the part of the geometric boundary $$S_\infty (\Sigma ^2)$$ of $$\Sigma ^2$$ where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of $$S_\infty (\Sigma ^2)$$ where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper (Cao and He, Infinitesimal rigidity of steady gradient Ricci soliton in dimension three. arXiv:1412.2714 , 2014, preprint), in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.

Intrinsic ultracontractivity via capacitary width

The semigroup associated with the Dirichlet heat kernel is said to be intrinsic ultracontractive if (i) the Dirichlet realization of the associated self adjoint operator has the first positive eigenvalue with positive L2 eigenfunction; (ii) the heat kernel is bounded above and below by the product of the eigenfunctions with positive multiplicative constants depending on time. We give an upper and lower estimate of the first eigenvalue in terms of capacitary width, which yields a sharp sufficient condition for (i). Our parabolic argument also yields an exponential decay property of a certain caloric measure. Then, the caloric measure is controlled by the elliptic Green function with the aid of a parabolic box argument. This is a crucial step for (ii). We give a sharp sufficient integral condition for intrinsic ultracontractivity in terms of capacitary width. A similar integral condition for the boundary Harnack principle is also obtained. Under geometric specifications, these integral conditions generalize known results and give more precise conditions. Sharpness is examined by an infinite funnel, for which we obtain a complete characterization of intrinsic ultracontractivity. Our method is purely analytic and elementary; it enables us to dispense with logarithmic Sobolev inequalities.

On the Schrödinger–Poisson system with a general indefinite nonlinearity

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: {−Δu+u+l(x)ϕu=k(x)g(u)+μh(x)uinR3,−Δϕ=l(x)u2inR3, where k∈C(R3) changes sign in R3, lim∣x∣→∞k(x)=k∞<0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that μ>μ1 and near μ1, where μ1 is the first eigenvalue of −Δ+id in H1(R3) with weight function h, whose corresponding positive eigenfunction is denoted by e1. An interesting phenomenon here is that we do not need the condition ∫R3k(x)e1pdx<0, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Costa and Tehrani, 2001).

Measure boundary value problems for semilinear elliptic equations with critical Hardy potentials

Let Ω⊂RN be a bounded C2 domain and Lκ=−Δ−κd2 where d=dist(.,∂Ω) and 0<κ≤14. Let α±=1±1−4κ, λκ the first eigenvalue of Lκ with corresponding positive eigenfunction ϕκ. If g is a continuous nondecreasing function satisfying ∫1∞(g(s)+|g(−s)|)s−22N−2+α+2N−4+α+ds<∞, then for any Radon measures ν∈Mϕκ(Ω) and μ∈M(∂Ω) there exists a unique weak solution to problem Pν,μ: Lκu+g(u)=ν in Ω, u=μ on ∂Ω. If g(r)=|r|q−1u (q>1), we prove that, in the supercritical range of q, a necessary and sufficient condition for solving P0,μ with μ>0 is that μ is absolutely continuous with respect to the capacity associated with the space B2−2+α+2q′,q′(RN−1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.

The limiting Behavior of Solutions to Inhomogeneous Eigenvalue Problems in Orlicz-Sobolev Spaces

Abstract The asymptotic behavior of the sequence {un} of positive first eigenfunctions for a class of inhomogeneous eigenvalue problems is studied in the setting of Orlicz-Sobolev spaces. After possibly extracting a subsequence, we prove that un → u∞ uniformly in Ω as n→∞, where u∞ is a nontrivial viscosity solution of a nonlinear PDE involving the ∞-Laplacian.

NONPARAMETRIC IDENTIFICATION OF POSITIVE EIGENFUNCTIONS

Important features of certain economic models may be revealed by studying positive eigenfunctions of appropriately chosen linear operators. Examples include long-run risk–return relationships in dynamic asset pricing models and components of marginal utility in external habit formation models. This paper provides identification conditions for positive eigenfunctions in nonparametric models. Identification is achieved if the operator satisfies two mild positivity conditions and a power compactness condition. Both existence and identification are achieved under a further nondegeneracy condition. The general results are applied to obtain new identification conditions for external habit formation models and for positive eigenfunctions of pricing operators in dynamic asset pricing models.

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

A condition for the existence of a minimum eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem.

Convexity properties of Dirichlet integrals and Picone-type inequalities

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains

Using three different notions of the generalized principal eigenvalue of linear second‐order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.

Ross Recovery in Continuous Time

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a positive semi-martingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C2 boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ1 and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\). We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ∂Ω.

Multiple positive solutions to a class of Kirchhoff equation on [formula omitted] with indefinite nonlinearity

We study the existence and multiplicity of positive solutions to a class of Kirchhoff equation −(1+b∫R3|∇u(x)|2dx)Δu(x)+u(x)=V(x)|u(x)|p−2u(x)+λk(x)u(x),u(x):=u∈H1(R3). When V(x) is sign changing in R3, we prove that there is δ̄>0 such that the Kirchhoff equation possesses at least one positive solution for 0<λ<λ1+δ̄ and at least two positive solutions for λ1<λ<λ1+δ̄. Here λ1 is the first eigenvalue of −Δu+u=λk(x)u, u∈H1(R3). A positive eigenfunction corresponding to λ1 is denoted by e1. An interesting phenomena is that the term b(∫R3|∇u|2dx)Δu can be used to compete with the indefinite nonlinearity and we do not assume the condition ∫R3V(x)e1pdx<0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (Alama and Tarantello, 1993).

The Robin eigenvalue problem for the [formula omitted]-Laplacian as [formula omitted

We study the asymptotic behavior, as p→∞, of the first eigenvalues and the corresponding eigenfunctions for the p(x)-Laplacian with Robin boundary conditions in an open, bounded domain Ω⊂RN with smooth boundary. We obtain uniform bounds for the sequence of first eigenvalues (suitably rescaled), and we prove that the positive first eigenfunctions converge uniformly in Ω to a viscosity solution of a problem involving the∞-Laplacian subject to appropriate boundary conditions.

Approximation complexity of tensor product-type random fields with heavy spectrum

We consider a sequence of Gaussian tensor product-type random fields Open image in new window , where Open image in new window and Open image in new window are all positive eigenvalues and eigenfunctions of the covariance operator of the process X1, Open image in new window are standard Gaussian random variables, and Open image in new window is a subset of positive integers. For each d ∈ ℕ, the sample paths of Xd almost surely belong to L2([0, 1]d) with norm ∥·∥2,d. The tuples Open image in new window, are the eigenpairs of the covariance operator of Xd. We approximate the random fields Xd, d ∈ Open image in new window, by the finite sums Xd(n) corresponding to the n maximal eigenvalues λk, Open image in new window.

Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems

We study the fully nonlinear parabolic equationF(D2um)−ut=0in Ω×(0,+∞),m⩾1, with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Ω⊂Rn, provided that the operator F is uniformly elliptic and positively homogeneous of order one. We prove that the renormalized limit of parabolic flow u(x,t) as t→+∞ is the corresponding positive eigenfunction which solvesF(D2φ)+μφp=0in Ω, where 0<p:=1m⩽1 and μ>0 is the corresponding eigenvalue. We also show that some geometric property of the positive initial data is preserved by the parabolic flow, under the additional assumptions that Ω is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, log(φ) is concave in the case p=1, and φ1−p2 is concave for 0<p<1.