Weighted Eigenvalue Problem Research Articles

Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics

We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape.First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in C1,1 sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the C1,α sense, for every α<1. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Spectral optimization for weighted anisotropic problems with Robin conditions

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains Ω⊂RN, N≥1, under Robin boundary conditions, proving the existence of two positive principal eigenvalues λ± respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of λ± with respect to the sign-changing weight, showing that the optimal eigenvalues Λ± are equal if the domain has a center of symmetry and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of Ω satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optimization problem for λ+ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in an heterogeneous habitat.

Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain Ω⊂RN, where the bang–bang weight equals a positive constant m¯ on a ball B⊂Ω and a negative constant −m̲ on Ω∖B. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher–KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of B in Ω. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of B vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from ∂Ω.

Linear theory for a mixed operator with Neumann conditions

We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in (Rev. Mat. Iberoam. 33(2) (2017), 377–416). Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in (Dipierro, Proietti Lippi and Valdinoci (2021)).

Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem \(-\Delta u =\lambda m u \) in a bounded smooth domain \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), under homogeneous Dirichlet boundary conditions, where \(\lambda \in \mathbb{R}\) and \(m\in L^\infty(\Omega)\). The domain \(\Omega\) represents the environment and \(m(x)\), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights \(m(x)\) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive.&#x0D; In our work we consider a sort of refinement of the result above. We write the weight \(m(x)\) as sum of two (or more) terms, i.e. \(m(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) and \(f_2(x)\) represent the spatial densities of the two resources which contribute to form the local growth rate \(m(x)\). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of \(f_1(x)\) and \(f_2(x)\) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if \(\Omega\) is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions.

A Weighted Eigenvalue Problems Driven by both p(·)-Harmonic and p(·)-Biharmonic Operators

Abstract The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u ∈ W 2 , p ( ⋅ ) ( Ω ) ∩ W 0 - 1 , p ( ⋅ ) ( Ω ) , \eqalign{&amp; \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left| u \right|^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr &amp; \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr} is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp (·)(Ω) and Wm,p (·)(Ω).

A weighted eigenvalue problem of the biased infinity Laplacian * *This work was partially supported by National Natural Science Foundation of China (No. 11501292 and 11531005).

We study a weighted eigenvalue problem of the β-biased infinity Laplacian operator arising from the β-biased tug-of-war. We characterize the principal eigenvalue by the comparison principle and show that β-biased infinity Laplacian operator possesses two principal eigenvalues, corresponding to a positive and a negative principal eigenfunction. When a parameter is less than the principal eigenvalue, certain existence and uniqueness results of the inhomogeneous equations related to this problem are established. As an application, we obtain the decay estimates for viscosity solutions of the parabolic problem associated to the β-biased infinity Laplacian. In the process, we also establish the Lipschitz regularity and Harnack inequality by barrier method.

A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian

In this paper, we study a weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian Δ∞hu+λa(x)|u|h−1u=0,inΩ,u=0,on∂Ω,where h>1,Δ∞hu=|Du|h−3Δ∞u is the h-homogeneous infinity Laplacian and a(x) is a positive continuous bounded function in Ω. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue, some existence and uniqueness results related to this problem are established.

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

AbstractLet $\Omega \subset \mathbb {R}^N$ , $N\geq 2$ , be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta )^s u =\lambda \rho u$ in $\Omega $ with homogeneous Dirichlet boundary condition, where $(-\Delta )^s$ , $s\in (0,1)$ , is the fractional Laplacian operator, $\lambda \in \mathbb {R}$ and $ \rho \in L^\infty (\Omega )$ .We study weak* continuity, convexity and Gâteaux differentiability of the map $\rho \mapsto 1/\lambda _1(\rho )$ , where $\lambda _1(\rho )$ is the first positive eigenvalue. Moreover, denoting by $\mathcal {G}(\rho _0)$ the class of rearrangements of $\rho _0$ , we prove the existence of a minimizer of $\lambda _1(\rho )$ when $\rho $ varies on $\mathcal {G}(\rho _0)$ . Finally, we show that, if $\Omega $ is Steiner symmetric, then every minimizer shares the same symmetry.

Strict Monotonicity and Unique Continuation for General Non-local Eigenvalue Problems

We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

In the present paper, we first consider the weighted eigenvalue problem $$\Delta _f u=\lambda _{f}u$$ in M with the Robin boundary condition $$\frac{\partial u}{\partial \nu }+\beta u=0$$ on $$\partial M$$ , where $$(M^n,g,e^{-f})$$ is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry–Emery Ricci curvature. We derive under some convexity condition of the boundary $$\partial M$$ , an explicit lower bound of the first weighted Robin eigenvalue $$\lambda _{1,f}(\beta )$$ depending only on the geometry of M and the constant $$\beta $$ appearing in the boundary condition. Another new estimate for $$\lambda _{1,f}(\beta )$$ with respect to the first nonzero Neumann eigenvalue $$\mu _{2,f}$$ of the weighted Laplacian $$\Delta _f$$ is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds.

Weighted eigenvalue problems involving a fourth-order elliptic equation with variable exponent

In this paper, we consider weighted fourth order nonhomogeneous elliptic problem with variable exponent. Under appropriate conditions, interval of a numerical parameter sufficiently small is derived for which the existence results are obtained. The proof relies on simple variational arguments based on Ekeland's variational principle.

Weighted eigenvalue problems involving a fourth-order elliptic equation with variable exponent

Weighted eigenvalue problems involving a fourth-order elliptic equation with variable exponent

SPECTRAL THEORETIC CHARACTERIZATION OF THE MASSLESS DIRAC ACTION

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator L. The corresponding counting function (number of eigenvalues between zero and a positive lambda) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as lambda tends to plus infinity and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem - metric, nonvanishing spinor field and topological charge - and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

An eigenvalue optimization problem for the p-Laplacian

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 &lt; p &lt; ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

Conformal spectral stability estimates for the Dirichlet Laplacian

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains inline image by reducing it, using conformal transformations, to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc inline image. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of “conformal regular” domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.

Weighted quasilinear eigenvalue problems in exterior domains

We consider the following weighted eigenvalue problem in the exterior domain: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _pu = \lambda &{}K(x) |u|^{p-2}u \quad \mathrm{in} \quad {B_1^c},\\ u = 0 &{}\quad \mathrm{on}\quad \partial B_1, \end{array}\right. } \end{aligned}$$ where \(\Delta _p\) is the \(p\)-Laplace operator with \(p>1,\) and \({B_1^c}\) is the exterior of the closed unit ball in \(\mathbb {R}^N\) with \(N\ge 1.\) There is no restriction on the dimension \(N\) in terms of \(p,\) i.e., we allow both \(1< p< N\) and \(p\ge N\). The weight function \(K\) is locally integrable on \({B_1^c}\) and is allowed to change its sign. For some appropriate choice of \(w\), a positive weight function on the interval \((1,\infty )\), we prove that the Beppo-Levi space \({{\mathcal D}^{1,p}_0(B_1^c)}\) is compactly embedded into the weighted Lebesgue space \(L^p({B_1^c};w(|x|)).\) The existence of the positive eigenvalue for the above problem is proved for \(K\) such that supp\(K^+\) is of non-zero measure and \( |K| \le w\). Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions.

On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in Axisymmetric Cold Plasmas

For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.

Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin–Dirichlet boundary conditions

We investigate the existence of principal eigenvalues type problems with weights for the quasilinear operator −Δp+Vψp with mixed weighted Robin–Dirichlet boundary conditions in a bounded regular domain. We also give some results on the existence of nonprincipal eigenvalues.