Dirichlet Eigenvalue Problem Research Articles

The Bézier curve and neural network model of the time-domain transient signals

The discussion is for the memory of Oleg Alexandrovich Tretyakov. In the discussion, one characteristic of the signal transfer of Professor Tretyakov's Evolutionary Approach to the Electromagnetics method is presented. Theoretically, the problem of the signal generated by a time-domain signal in a waveguide is addressed. The theoretic propagation is realized-exemplified through the actual TEC (TECU) map estimating by the Bézier curve and neural network. The striking aspect of the segmented prototype established with the Bézier approach is its adaptability. This mechanical curve, which does not need any preliminary preparation, is framed on differential geometric invariants. In the essay, a time-dependent complete set of magnetic waveguide modes is remembered. While the Dirichlet and Neumann eigenvalue problems determine the vector functions of the modes, the behavior of the time-evolving amplitudes is given by the Klein-Gordon equation. Examples of current data are discussed with the TEC map for the 2017 year. While the actual fluctuation of the interpolated CODE TEC atlas is illustrated, the mechanical Bézier curve family (untouched before) and the neural network introduce time-domain estimations to the reader. The parametric curve approach governs the Bézier model. The curve, which is C0 class segmented continuously, models on its way with new hourly components every twelve hours. The network model employs the solar wind parameters for the TEC atlas estimation. The reliability and consistency of the models are exhibited by the R correlation ratio, absolute error, and mean squared error. As a result, the R coefficient of the curve and network models vary around 91.2% and 98.8%, respectively. One can note the error of the network model falls to 1.1308 TECU. The outcomes are compatible with the former discussions.

Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting II

We study positive solutions of a logistic elliptic equation with a nonlinear boundary condition that models coastal fishery harvesting ([18]). An essential role is played by the smallest eigenvalue of the Dirichlet eigenvalue problem, in terms of which, a noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case. We also further study the noncritical case for a more precise description of the positive solution set, including uniqueness and stability analysis for large parameters. Our approach relies on an energy method, sub- and supersolutions, and implicit function analysis.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

Spectral convergence of high-dimensional spheres to Gaussian spaces

We prove that the spectral structure on the N -dimensional standard sphere of radius (N - 1)^{1/2} compatible with a projection onto the first n -coordinates converges to the spectral structure on the n -dimensional Gaussian space with variance 1 as N \rightarrow \infty . We also show the analogue for the first Dirichlet eigenvalue problem on a ball in the sphere and that on a halfspace in the Gaussian space.

Eigenvalue problems for p-div-curl systems

We consider eigenvalue problems for a general p-div-curl system. In particular, we consider a nonlinear eigenvalue problem similar to the standard Dirichlet eigenvalue problem for the p-Laplacian, but due to the presence of the curl, we see that eigenfunctions of the general p-div-curl operator do not enjoy nice regularity as in the p-Laplacian case. We first show that weak solutions of a particular nonlinear p-div-curl eigenvalue problem split into a good and bad part, where the good part is not even divergence free. We then show that the general p-div-curl system admits a Ljusternik-Schnirelman sequence of eigenvalues. Finally, we show a general chain rule for the curl and divergence.

A Faber-Krahn inequality for mixed local and nonlocal operators

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.

Regular convergence and finite element methods for eigenvalue problems

Regular convergence, together with other types of convergence, have been studied since the 1970s for discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of a compact operator $T$ that can be written as an eigenvalue problem of a holomorphic Fredholm operator function $F(\eta) = T-\frac{1}{\eta} I$. Focusing on finite element methods (conforming, discontinuous Galerkin, non-conforming, etc.), we show that the regular convergence of the discrete holomorphic operator functions $F_n$ to $F$ follows from the compact convergence of the discrete operators $T_n$ to $T$. The convergence of the eigenvalues is then obtained using abstract approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to prove the convergence of various finite element methods for eigenvalue problems such as the Dirichlet eigenvalue problem and the biharmonic eigenvalue problem.

Risk-sensitive control for a class of diffusions with jumps

We consider a class of diffusions controlled through the drift and jump size, and driven by a jump Lévy process and a nondegenerate Wiener process, and we study infinite horizon (ergodic) risk-sensitive control problems for this model. We start with the controlled Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results in the literature on exit rate control problems. Then we consider the infinite horizon average risk-sensitive minimization and maximization problems on the whole domain. Under suitable hypotheses, we establish existence and uniqueness of a principal eigenfunction for the Hamilton–Jacobi–Bellman (HJB) operator on the whole space, and fully characterize stationary Markov optimal controls as the measurable selectors of this HJB equation.

Principal eigenvalue problem for infinity Laplacian in metric spaces

Abstract This article is concerned with the Dirichlet eigenvalue problem associated with the ∞ \infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞ \infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for p p -Laplacian in the Euclidean space.

Weighted local Weyl laws for elliptic operators

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\man$ of dimension $n>0$, formally positive self-ajdoint with respect to some positive smooth density $d\mu_\man$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\R$ and define \ \[ K_L^s(x,y)=\sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}\, . \] We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by H\ormander in [Hor68]. In the present work we build on H\ormander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$. If $s=\frac{n}{m}$, under some explicit generic condition on the principal symbol of $A$, which holds if $A$ is a differential operator, the kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.

Upper bound estimates of eigenvalues for Hörmander operators on non-equiregular sub-Riemannian manifolds

We study the closed eigenvalue problem and Dirichlet eigenvalue problem of self-adjoint Hörmander operators on non-equiregular sub-Riemannian manifolds. By Rayleigh-Ritz formula and the subelliptic heat kernel estimates, we establish the upper bounds of eigenvalues which depend on the volume of the subunit ball and the measure of the manifold. Under a certain condition, we obtain the explicit upper bounds of eigenvalues which have the polynomially growth in k with the optimal order related to the non-isotropic dimension of the manifold.

On the Robin spectrum for the equilateral triangle* *Dedicated to Michael Berry for his 80th birthday.

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin–Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.

Covariance of error terms related to the Dirichlet eigenvalue problem

We explore the covariance of error terms coming from Weyl's conjecture regarding the number of Dirichlet eigenvalues up to size X. We also consider this problem in short intervals, that is, the error term of the number of eigenvalues in the window [ X , X + S ] $[X, X+S]$ for some S ( X ) $S(X)$ . We look at these error terms for planar domains where the Dirichlet eigenvalues can be explicitly calculated. In these cases, the error term is closely related to the error term from the classical lattice points counting problem of expanding planar domains. We give a formula for the covariance of such error terms, for general planar domains. We also give a formula for the covariance of error terms in short intervals, for sufficiently large intervals. Going back to the Dirichlet eigenvalue problem, we give results regarding the covariance of the error terms in short intervals of “generic” rectangles. We also explore a specific example, namely we compute the covariance between the error terms of an equilateral triangle and various rectangles.

Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (-Delta )^s in bounded open Lipschitz sets in the small order limit s rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 0^+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log |xi |. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L^2-normalized Dirichlet eigenfunctions of (-Delta )^s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L^2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method

This paper introduces a novel hybrid high-order (HHO) method to approximate the eigenvalues of a symmetric compact differential operator. The HHO method combines two gradient reconstruction operators by means of a parameter 0<alpha <~1 and introduces a novel cell-based stabilization operator weighted by a parameter 0<beta <infty . Sufficient conditions on the parameters alpha and beta are identified leading to a guaranteed lower bound property for the discrete eigenvalues. Moreover optimal convergence rates are established. Numerical studies for the Dirichlet eigenvalue problem of the Laplacian provide evidence for the superiority of the new lower eigenvalue bounds compared to previously available bounds.

A Convergence Algorithm of Boundary Elements for the Laplace Operator's Dirichlet Eigenvalue Problem

A partial differential equation has been using the various boundary elements techniques for getting the solution to eigenvalue problem. A number of mathematical concepts were enlightened in this paper in relation with eigenvalue problem. Initially, we studied the basic approaches such as Dirichlet distribution, Dirichlet process and the Model of mixed Dirichlet. Four different eigenvalue problems were summarized, viz. Dirichlet eigenvalue problems, Neumann eigenvalue problems, Mixed Dirichlet-Neumann eigenvalue problem and periodic eigenvalue problem. Dirichlet eigenvalue problem was analyzed briefly for three different cases of value of λ. We put the result for multinomial as its prior is Dirichlet distribution. The result of eigenvalues for the ordinary differential equation was extrapolated. The Basic mathematics was also performed for λ calculations which follow iterative method.

A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The method removes the outliers appearing in the high-frequency region of the approximate spectrum when using the Cp−1,p-th (p≥3) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multidimensional problems. To remove the outliers, we penalize the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra-computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis of the Neumann eigenvalue problem (for p≥2). The boundary penalization does not change the test and solution spaces. In the limiting case, when the penalty goes to infinity, we perform the dispersion analysis of C2 cubic elements for the Dirichlet eigenvalue problem and C1 quadratic elements for the Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.

Morse index versus radial symmetry for fractional Dirichlet problems

In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions u to the semilinear fractional Dirichlet problem(−Δ)su=f(u) in B,u=0in RN∖B, where s∈(0,1), B⊂RN is the unit ball centered at zero and the nonlinearity f is of class C1. We prove that for s∈(1/2,1) any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to N+1. If s∈(0,1/2], the same conclusion holds under an additional assumption on f. In particular, our results apply to the Dirichlet eigenvalue problem for the operator (−Δ)s in B for all s∈(0,1), and imply that eigenfunctions corresponding to the second Dirichlet eigenvalue in B are antisymmetric. This resolves a conjecture of Bañuelos and Kulczycki.

Formula omitted] bounds of eigenfunctions with an indefinite weight function

This paper is concerned with the L∞ bounds of eigenfunctions to the eigenvalue problem Δu=−λm(x)uin Ω,u=0on ∂Ω,where Ω is a bounded domain and m(x) is an indefinite weight function. For the Dirichlet eigenvalue problem when m(x)≡1, the L∞ bound of eigenfunction is obtained by classical heat kernel estimate. However, the case is quite different for the weighted problem, additional difficulty appears if m(x) is indefinite. In this paper, we investigate the L∞ bounds of eigenfunctions by the means of comparison argument and fundamental solution of heat equation.

A new solution to the conformal Skorokhod embedding problem and applications to the Dirichlet eigenvalue problem

In a recent work by Gross, the following problem was posed and solved: given a measure μ on R with finite second moment, find a simply connected domain U in C such that the real part of the random variable ZτU has the distribution μ, where Z is a planar Brownian motion and τU is the exit time from U. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show, following a method due to Mariano and Panzo, that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution μ. This gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.