Approximation Of Green's Function Research Articles

The time‐asymptotic expansion for the compressible Euler equations with damping

AbstractIn 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time‐asymptotically converges to the diffusion wave of the porous media equation. Geng et al. proposed a time‐asymptotic expansion around the diffusion wave , which is a better asymptotic profile than . In this paper, we rigorously justify the time‐asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time‐asymptotic expansion.

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

Let Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} be a bounded domain in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^N$$\\end{document} with C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} boundary and let K⊂∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\subset \\partial \\Omega $$\\end{document} be either a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} submanifold of the boundary of codimension k<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k<N$$\\end{document} or a point. In this article we study various problems related to the Schrödinger operator Lμ=-Δ-μdK-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{\\mu } =-\\Delta - \\mu d_K^{-2}$$\\end{document} where dK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_K$$\\end{document} denotes the distance to K and μ≤k2/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\le k^2/4$$\\end{document}. We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range μ≤k2/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\le k^2/4$$\\end{document}.

Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

The goal of this paper is to establish Green function estimates for a class of purely discontinuous symmetric Markov processes with jump kernels degenerate at the boundary and critical killing potentials. The jump kernel and the killing potential depend on several parameters. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of the parameters involved. Depending on the regions where the parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Together with the main results of our previous paper [Potential Anal., online, 2021], we completely determine the region of the parameters where the boundary Harnack principle holds.

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions of generalized porous medium equations of the form∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L.In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques.We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator.Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.

Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

In this paper we consider the Dirichlet form on the half-space \({\mathbb R}^d_+\) defined by the jump kernel \(J(x,y)=|x-y|^{-d-\alpha }{{\mathcal {B}}}(x,y)\), where \({{\mathcal {B}}}(x,y)\) can be degenerate at the boundary. Unlike our previous works [16, 17] where we imposed critical killing, here we assume that the killing potential is identically zero. In case \(\alpha \in (1,2)\) we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored \(\alpha \)-stable process, \(\alpha \in (1,2)\), in the half-space studied in [2].

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

Validation of the Green's Function Approximation for the Calculation of Magnetic Exchange Couplings.

In this work, we assess the potential of the Green's function approximation to predict isotropic magnetic exchange couplings and to reproduce the standard broken-symmetry energy difference approach for transition metal complexes. To this end, we have selected a variety of heterodinuclear, homodinuclear, and polynuclear systems containing 3d transition metal centers and computed the couplings using both the Green's function and energy difference methods. The Green's function approach is shown to have mixed results for the cases tested. For dinuclear complexes with large strength couplings (≳50 cm-1), the Green's function method is unable to reliably reproduce the energy difference values. However, for weaker dinuclear couplings, the Green's function approach acceptably reproduces broken-symmetry energy difference couplings. In polynuclear cases, the Green's function approximation worked remarkably well, especially for FeIII complexes. On the other hand, for a NiII polynuclear complex, qualitatively wrong couplings are predicted. Overall, the evaluation of exchange couplings from local rigid magnetization rotations offers a powerful alternative to time-consuming energy differences methods for large polynuclear transition metal complexes, but to achieve a quantitative agreement, some improvements to the method are needed.

Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition

In the context of hyperbolic systems of balance laws, the Shizuta–Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition (SK) in 3D, such that some eigendirections do not exhibit dissipation at all. We prove that if the source term is nonresonant (in a suitable sense) in the direction where dissipation does not play any role, then the formation of singularities is prevented, despite the lack of dissipation, and the smooth solutions exist globally in time. The main idea of the proof is to couple Green function estimates for weakly dissipative hyperbolic systems with the space–time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah. More precisely, the partially dissipative hyperbolic systems violating (SK) are endowed, in the nondissipative directions, with a special structure of the nonlinearity, the so-called nonresonant bilinear form for the wave equation (see Pusateri and Shatah, CPAM 2013).

Boundary Harnack principle for diffusion with jumps

Consider the operator Lb=L0+b1⋅∇+Sb2 on Rd, where L0 is a second order differential operator of non-divergence form, the drift function b1 belongs to some Kato class and Sb2f(x)≔∫Rdf(x+z)−f(x)−∇f(x)⋅z1{|z|≤1}b2(x,z)j0(z)dz,f∈Cb2(Rd).Here j0(z) is a nonnegative locally bounded function on Rd∖{0} satisfying that ∫Rd(1∧|z|2)j0(z)dz<∞ and that there are constants β∈(1,2) and c0>0 so that j0(z)≤c0|z|d+βfor|z|≤1,and b2(x,z) is a real-valued bounded function on Rd×Rd. There is conservative Feller process Xb associated with the non-local operator Lb. We derive sharp two-sided Green function estimates of Lb on bounded C1,1 domains, identify the Martin and minimal Martin boundary, and establish the Martin integral representation of Lb-harmonic functions on these domains. The latter in particular reveals how the process Xb exits a bounded C1,1 domain D, or equivalently, the structure of the harmonic measure of Lb on D, which consists of the continuously exiting term and the jump-off term. These results are then used to establish, under some mild conditions, Harnack principle and the boundary Harnack principle with explicit boundary decay rate for the operator Lb on C1,1 open sets.

Approximate time–energy uncertainty relationship from the fixed-energy sum over paths approach

In this article, we show how an approximate time–energy uncertainty relationship of the form Δ EΔ t ≃ ℏ can be derived in the context of the fixed-energy sum over paths approach to quantum bound systems. The relationship connects the indeterminacy Δ t on the travel time of the quantum object to the width Δ E of the resonance in the approximate Green function corresponding to an allowed value of energy. The mathematical origin of the relationship is to be tracked to the Fourier transform relationship between the time propagator and energy-dependent Green function; however, the core of the derivation does not use advanced mathematics, may be carried out using mostly graphical and geometrical methods, and may provide insight to students on the meaning and origin of the time–energy uncertainty relationship. Our work may contribute to close a gap by which the time–energy relationship is most often taught with insufficient explanation at elementary level.

On the Retrieval of Forward-Scattered Waveforms From Acoustic Reflection and Transmission Data With the Marchenko Equation.

A Green's function in an acoustic medium can be retrieved from reflection data by solving a multidimensional Marchenko equation. This procedure requires a priori knowledge of the initial focusing function, which can be interpreted as the inverse of a transmitted wavefield as it would propagate through the medium, excluding (multiply) reflected waveforms. In practice, the initial focusing function is often replaced by a time-reversed direct wave, which is computed with help of a macro velocity model. Green's functions that are retrieved under this (direct-wave) approximation typically lack forward-scattered waveforms and their associated multiple reflections. We examine whether this problem can be mitigated by incorporating transmission data. Based on these transmission data, we derive an auxiliary equation for the forward-scattered components of the initial focusing function. We demonstrate that this equation can be solved in an acoustic medium with mass density contrast and constant propagation velocity. By solving the auxiliary and Marchenko equation successively, we can include forward-scattered waveforms in our Green's function estimates, as we demonstrate with a numerical example.

Nonlocal scalar fields in static spacetimes via heat kernels

We solve the non-local equation ${-e^{-\ell^2\square}\square{\phi}=J}$ for i) static scalar fields in static spacetimes and ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as non-local Poisson/inhomogeneous Helmholtz equations in compact and non-compact weighted/Riemannian manifolds using static/frequency-domain Green's functions, which can be computed from the heat kernels in the respective manifolds. With the help of the heat kernel estimates, we derive the static Green's function estimates and use them to discuss the regularity. We also present several examples of exact and estimated static/frequency-domain Green's functions.

Green function estimates on complements of low-dimensional uniformly rectifiable sets

It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions. arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators L_{beta ,gamma } =- {text {div}}D^{d+1+gamma -n} nabla associated to a domain Omega subset {mathbb {R}}^n with a uniformly rectifiable boundary Gamma of dimension d < n-1, the now usual distance to the boundary D = D_beta given by D_beta (X)^{-beta } = int _{Gamma } |X-y|^{-d-beta } dsigma (y) for X in Omega , where beta >0 and gamma in (-1,1). In this paper we show that the Green function G for L_{beta ,gamma }, with pole at infinity, is well approximated by multiples of D^{1-gamma }, in the sense that the function big | Dnabla big (ln big ( frac{G}{D^{1-gamma }} big )big )big |^2 satisfies a Carleson measure estimate on Omega . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical" distance function from David et al. (Duke Math J, to appear).

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Helgason-Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we prove in a unified way that the sharp constant in the n−12-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n−12-th order Sobolev constant when n is odd and n≥5. We will also establish a lower bound of the coefficient of the Hardy term for the k-th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension n and k-th order derivatives. As a consequence, we thus show that the sharp constant in the k-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n is strictly less than the best k-th order Sobolev constant for all n≥2k+2. Precise expressions and optimal bounds for Green's functions of the operator −ΔH−(n−1)24 on the hyperbolic space Bn and operators of the product form are given, where (n−1)24 is the spectral gap for the Laplacian −ΔH on Bn. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space in terms of hypergeometric functions. In fact, we will establish the precise formulas of the Green's functions of the operator∏j=0k−1((ν+j)2−(n−1)2/4−ΔH),ν≥0, in terms of hypergeometric function F(a,b;c,z).Our approach to establish the main theorems is using the Helgason-Fourier analysis on hyperbolic spaces and are substantially different from that in dealing with the first order Hardy-Sobolev-Maz'ya inequalities on upper half spaces.

Approximate Atomic Green Functions.

In atomic and many-particle physics, Green functions often occur as propagators to formally represent the (integration over the) complete spectrum of the underlying Hamiltonian. However, while these functions are very crucial to describing many second- and higher-order perturbation processes, they have hardly been considered and classified for complex atoms. Here, we show how relativistic (many-electron) Green functions can be approximated and systematically improved for few- and many-electron atoms and ions. The representation of these functions is based on classes of virtual excitations, or so-called excitation schemes, with regard to given bound-state reference configurations, and by applying a multi-configuration Dirac-Hartree-Fock expansion of all atomic states involved. A first implementation of these approximate Green functions has been realized in the framework of Jac, the Jena Atomic Calculator, and will facilitate the study of various multi-photon and/or multiple electron (emission) processes.

Global Green's function estimates for the convection–diffusion equation

For the stationary convection–diffusion equation in , n>2, we give a sufficient condition on the bounded non-compactly supported drift that guarantees the existence of a fundamental solution with global two-sided estimates of the Newtonian potential type.

On the Green's function emergence from interferometry of seismic wave fields generated in high-melt glaciers: implications for passive imaging and monitoring

Abstract. Ambient noise seismology has revolutionized seismic characterization of the Earth's crust from local to global scales. The estimate of Green's function (GF) between two receivers, representing the impulse response of elastic media, can be reconstructed via cross-correlation of the ambient noise seismograms. A homogenized wave field illuminating the propagation medium in all directions is a prerequisite for obtaining an accurate GF. For seismic data recorded on glaciers, this condition imposes strong limitations on GF convergence because of minimal seismic scattering in homogeneous ice and limitations in network coverage. We address this difficulty by investigating three patterns of seismic wave fields: a favorable distribution of icequakes and noise sources recorded on a dense array of 98 sensors on Glacier d'Argentière (France), a dominant noise source constituted by a moulin within a smaller seismic array on the Greenland Ice Sheet, and crevasse-generated scattering at Gornergletscher (Switzerland). In Glacier d'Argentière, surface melt routing through englacial channels produces turbulent water flow, creating sustained ambient seismic sources and thus favorable conditions for GF estimates. Analysis of the cross-correlation functions reveals non-equally distributed noise sources outside and within the recording network. The dense sampling of sensors allows for spatial averaging and accurate GF estimates when stacked on lines of receivers. The averaged GFs contain high-frequency (&gt;30 Hz) direct and refracted P waves in addition to the fundamental mode of dispersive Rayleigh waves above 1 Hz. From seismic velocity measurements, we invert bed properties and depth profiles and map seismic anisotropy, which is likely introduced by crevassing. In Greenland, we employ an advanced preprocessing scheme which includes match-field processing and eigenspectral equalization of the cross spectra to remove the moulin source signature and reduce the effect of inhomogeneous wave fields on the GFs. At Gornergletscher, cross-correlations of icequake coda waves show evidence for homogenized incident directions of the scattered wave field. Optimization of coda correlation windows via a Bayesian inversion based on the GF cross coherency and symmetry further promotes the GF estimate convergence. This study presents new processing schemes on suitable array geometries for passive seismic imaging and monitoring of glaciers and ice sheets.