Elliptic Operators Research Articles

Study of weighted elliptic composition operators on the unit ball of ℂ N

Abstract We study the general properties, point spectrum and spectrum of a weighted composition operator W m , φ {W}_{m,\varphi } with elliptic symbol φ \varphi on the unit ball B N {{\mathbb{B}}}_{N} of C N {{\mathbb{C}}}^{N} , and general weight m ∈ Hol ( B N ) m\in {\rm{Hol}}\left({{\mathbb{B}}}_{N}) . We give a complete description of the spectra in the majority of cases, and we provide inclusions in general.

Monotonicity results of solutions to the uniformly elliptic nonlocal Bellman system

In this paper, we consider the uniformly elliptic nonlocal Bellman problem Fsu(x)=f(u(x),v(x)),Fsv(x)=g(u(x),v(x)).Firstly, we study narrow region principles for the uniformly elliptic nonlocal Bellman operators in bounded and unbounded domains, which play key roles in obtaining the main results by the process of sliding method. Then we deal with monotonicity properties of solutions to the uniformly elliptic nonlocal Bellman system.

Convergence analysis of enhanced Phragmén–Lindelöf methods for solving elliptic Dirichlet problems

In this paper, we explore the ball convergence properties of enhanced Phragmén–Lindelöf type methods for solving the Dirichlet problem with an elliptic operator. By placing requirements on the elliptic operator and auxiliary points, we characterize the ball limit of a series of elliptic operators in non‐divergence form. Considering all dimensions, it becomes apparent that dealing solely with measurable and bounded coefficients for this class of operators is insufficient, necessitating additional regularity assumptions on them. We find the radii of convergence balls using recurrence relations, so as to guarantee the convergence of iterative reconstruction techniques, beginning from any point inside the ball centered on an auxiliary point.

A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator

A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator

High-frequency homogenization of multidimensional hyperbolic equations

In L 2 ( R d ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − div ⁡ g ( x / ϵ ) ∇ + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For hyperbolic equations with the operator A ϵ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator A 1 are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 ( R d ) -norm for small ε are obtained.

The spectral determinant for second-order elliptic operators on the real line

The spectral determinant for second-order elliptic operators on the real line

Neumann eigenvalues of elliptic operators in Sobolev extension domains

We obtain estimates of Neumann eigenvalues of the divergence form elliptic operators in Sobolev extension domains. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. The connection between Neumann eigenvalues of elliptic operators and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857) is given.

Boundary triples for a family of degenerate elliptic operators of Keldysh type

Boundary triples for a family of degenerate elliptic operators of Keldysh type

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

Nonconforming virtual element methods for velocity-pressure Stokes eigenvalue problem

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we “enhance” the definition of the virtual element space to make the L2-projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance.

Uniqueness of second-order elliptic operators with unbounded and degenerate coefficients in L 1-spaces

Let . Let and . Consider the formal second-order differential operator in . We show that the closure of is quasi-m-accretive under certain conditions on the coefficients.

Symmetrization for Mixed Operators

Abstract In this paper, we prove Talenti’s comparison theorem for mixed local/nonlocal elliptic operators and derive the Faber–Krahn inequality for the first eigenvalue of the Dirichlet mixed local/nonlocal problem. Our findings are relevant to the fractional p&q−Laplacian operator.

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Numerical integrator for highly oscillatory differential equations based on the Neumann series

AbstractWe propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

On a new singular and degenerate extension of the [formula omitted]-Laplace operator

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.

Fractional Elliptic Operators with Multiple Poles on Riemannian Manifold with Clifford Bundle

Fractional Elliptic Operators with Multiple Poles on Riemannian Manifold with Clifford Bundle

Method of fundamental solutions: New approximation results and applications

The approximation properties of shifted fundamental solutions are stated for a general class of second order linear elliptic differential operators. Motivated by the method of the fundamental solutions, the singularities of the fundamental solutions are located in the vicinity of a fixed surface, which is the boundary of a 3-dimensional Lipschitz domain. The approximation is estimated first in Ck-norm. The corresponding convergence rate is determined depending on the density of the singularities. This delivers a solid theoretical basis for the method of fundamental solutions. The results are extended to exterior domains and as a new application, the numerical solution of Dirichlet-to-Neumann problems are constructed and analyzed on unbounded domains. The efficiency of this new method is demonstrated in series of numerical experiments.

Qualitative properties of solutions for dual parabolic equation involving uniformly elliptic nonlocal operator

In this research, we study certain characteristics of the solution for an equation involving uniformly elliptic nonlocal operator by establishing various maximum principles in bounded and unbounded regions. Additionally, we don't need the conditions that the narrow region principle's bound assumption and decay in the unbounded domain. In order to get the monotonicity of the solution, we use several maximum principles and averaging effects to overcome the challenges that generated by the uniformly elliptic nonlocal operator and fractional‐order variable . Of course, while investigating the qualitative characteristics of the equation, we also provide some estimates of auxiliary function. For the study of additional nonlocal operators, we believe the method of uniformly elliptic nonlocal operators will also be helpful.

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty }$ -manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell }_{p}$ with $p>2$ or in the spaces $C^{\ell }$ (specifically, for the $p$ -th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.