Poisson Operator Research Articles

Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive mathcal {R}-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

On Lip(ω)-Continuity of the Operator of Harmonic Reflection Over Boundaries of Simple Carathéodory Domains

We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Caratheodory domains. This operator is considered as one acting from a space of functions of Lipschitz–Holder type defined by a general modulus of a continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question. This criterion is also obtained in the paper. These results generalize and refine those obtained in the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019).

Statistical mechanics with non-integrable topological constraints: Self-organization in knotted phase space

The object of this study is the statistical mechanics of dynamical systems lacking a Hamiltonian structure due to the presence of non-integrable topological constraints that limit the accessible regions of the phase space. Focusing on the simplest three dimensional case, we develop a procedure (Poissonization) that assigns to any three dimensional non-Hamiltonian system an equivalent four dimensional Hamiltonian system endowed with a proper time. The statistical distribution is then constructed in the recovered four dimensional canonical phase space. Projecting in the original reference frame, we show that the statistical distribution departs from standard Maxwell–Boltzmann statistics. The deviation is a function of the knottedness of the phase space, which is measured by the helicity density of the topological constraint. The theory is then generalized to a class of non-Hamiltonian systems in higher dimensions.

SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION FOR THE NONLOCAL POISSON EQUATION

In this paper, we study new classes of boundary value problems for a nonlocal analogue of the Poisson equation. The boundary conditions, as well as the nonlocal Poisson operator, are specified using transformation operators with orthogonal matrices. The paper investigates the questions of solvability of analogues of boundary value problems of the Dirichlet and Neumann type. It is proved that, as in the classical case, the analogue of the Dirichlet problem is unconditionally solvable. For it, theorems on the existence and uniqueness of the solution to the problem are proved. An explicit form of the Green's function, a generalized Poisson kernel, and an integral representation of the solution are found. For an analogue of the Neumann problem, an exact solvability condition is found in the form of a connection between integrals of given functions. The Green's function and an integral representation of the solution of the problem under study are also constructed.

On -reflection of harmonic functions and -approximation by harmonic polynomials

We obtain several new sharp -continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in . These results are based on a new criterion (also obtained in this paper) for -continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for -approximability of functions by harmonic polynomials on boundaries of simple Carathéodory domains in . Bibliography: 17 titles.

Dissipative brackets for the Fokker–Planck equation in Hamiltonian systems and characterization of metriplectic manifolds

It is shown that the Fokker–Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic bracket formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth. First, a microscopic metriplectic bracket is derived for the stochastic equations of motion that characterize the random walk of the elements constituting the statistical ensemble. Such bracket is fully determined by the Poisson operator generating the Hamiltonian dynamics of an isolated (unperturbed) particle. Then, the macroscopic metriplectic bracket associated with the evolution of the distribution function of the ensemble is induced from the microscopic metriplectic bracket. Similarly, the macroscopic Casimir invariants are inherited from microscopic dynamics. The theory is applied to construct the Fokker–Planck equation of an infinite dimensional Hamiltonian system, the Charney–Hasegawa–Mima equation. Finally, the canonical form of the symmetric (dissipative) part of the metriplectic bracket is identified in terms of a ‘canonical metric tensor’ corresponding to an Euclidean metric tensor on the symplectic leaves foliated by the Casimir invariants.

The two-weight inequality for the Poisson operator in the Bessel setting

Fix λ>0. Consider the Bessel operator Δλ:=−d2dx2−2λxddx on R+:=(0,∞) and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator Pt[λ]=e−tΔλ in this Bessel setting. In particular, we prove that for a measure μ on R+,+2:=(0,∞)×(0,∞) and σ on R+:‖Pσ[λ](f)‖L2(R+,+2;μ)≲‖f‖L2(R+;σ), if and only if testing conditions hold for the Poisson operator and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in the testing conditions.

On The Approximation of Functions from The Hölder Class Given On a Segment by Their Biharmonic Poisson Operators

We obtain the exact equality for the upper bounds of deviations of biharmonic Poisson operators on the Holder classes of functions continuous on the segment [−1; 1].

Beltrami operators and their application to constrained diffusion in Beltrami fields

Beltrami fields occur as stationary solutions of the Euler equations of fluid flow and as force free magnetic fields in magnetohydrodynamics. In this paper we discuss the role of Beltrami fields when considered as operators acting on a Hamiltonian function to generate particle dynamics. Beltrami operators, which include Poisson operators as a special subclass, arise in the description of topologically constrained diffusion in non-Hamiltonian systems. Extending previous results (Sato and Yoshida 2018 Phys. Rev. E 97 022145), we show that random motion generated by a Beltrami operator satisfies an H-theorem, leading to a generalized Boltzmann distribution on the coordinate system where the Beltrami condition holds. When the Beltrami condition is violated, random fluctuations do not work anymore to homogenize the particle distribution in the coordinate system where they are applied. The resulting distribution becomes heterogeneous. The heterogeneity depends on the ‘field charge’ measuring the departure of the operator from a Beltrami field. Examples of both Beltrami and non-Beltrami operators in three and four real dimensions together with the corresponding equilibrium distribution functions are given.

On $${{\,\mathrm{\textit{Lip}}\,}}^m$$ Lip m -reflection of harmonic functions over boundaries of simple Carathéodory domains

We obtain several new sharp necessary and sufficient $${{\,\mathrm{\textit{Lip}}\,}}^m$$ -continuity conditions for operators of harmonic reflection of functions over boundaries of simple Caratheodory domains in $${\mathbb {R}}^N$$ . These results are based on our $${{\,\mathrm{\textit{Lip}}\,}}^m$$ -continuity criterion for the Poisson operator in the aforementioned domains.

Poisson type operators on the Fock space of type B

The main purpose of this paper is to propose an (α, q)-analogue of the Poisson operators on the Fock space of type B in the sense of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] and to find a probability law of this operator. We shall show that the probability law is expressed by the q-Meixner distribution in the sense of Definition 3.2. Our results contain not only symmetric distributions as in Bożejko-Ejsmont-Hasebe but also the non-symmetric ones such as free Poisson, q and q2-deformations of Poisson, Pascal, Gamma, and Meixner distributions.

On the stability and accuracy of partially and fully implicit schemes for phase field modeling

We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen–Cahn and Cahn–Hilliard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations:1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy;2. most implicit schemes are energy-stable if the time-step size is sufficiently small;3. a traditionally known conditionally energy-stable scheme still possesses an unconditionally energy-stable physical solution;4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough;5. a first-order FIS for the Allen–Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies |uh(x)|≤1 for all x and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

Divergence Theorem in the L2-Version. Application to the Dirichlet Problem

We propose an L2 -version of the divergence theorem. The Green and Poisson operators associated with the infinite-dimensional version of the Dirichlet problem are investigated.

The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

Abstract This is the 3rd paper in a series [6, 9] analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix $S_h$, at some fixed energy $E$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ that are perturbations of the free Hamiltonian $h^2 \Delta $ on $L^2(\mathbb{R}^d)$ by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha } v(\hat x)$ as $x \to \infty $, $\hat x = x/|x|$, for some $\alpha> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $\mu $ on $\mathbb{S}^1$. Moreover, $\mu $ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results in [14] on the classical Hamiltonian dynamics and semiclassical Poisson operator to the larger class of potentials under consideration here.

Exponential instability in the fractional Calderón problem

In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this result, we follow the strategy introduced by Mandache in (2001 Inverse Problems 17 1435) for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in Rüland and Salo (2017 arXiv:1708.06294). Finally, in one dimension, we show a close relation between the fractional Calderón problem and the truncated Hilbert transform.

The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.

Direct and Inverse Theorems on the Approximation of 2π-Periodic Functions by Taylor–Abel–Poisson Operators

We prove direct and inverse theorems on the approximation of 2π -periodic functions by Taylor–Abel–Poisson operators in the integral metric.

Approximating Properties of Biharmonic Poisson Operators in the Classes L ̂ β , 1 ψ $$ {\widehat{L}}_{\beta, 1}^{\psi } $$

We obtain asymptotic equalities for the least upper bounds of the approximations of functions from theclasses $$ {\widehat{L}}_{\beta, 1}^{\psi } $$ by biharmonic Poisson operators in the integral metric.

Hardy space estimates of Hausdorff operators on the Heisenberg group

This is a continuation of a series of works about Hausdorff operator on Hardy spaces in the setting of the Heisenberg group. In this paper, we obtain the boundedness from Lebesgue spaces to Hardy spaces for fractional Hausdorff operators and their compositions with Riesz transforms. We also establish the boundedness for two kinds of special Hausdorff operators, the Hausdorff–Poisson operator and the Hausdorff–Gauss operator, on Hardy spaces.

Power diagrams and sparse paged grids for high resolution adaptive liquids

We present an efficient and scalable octree-inspired fluid simulation framework with the flexibility to leverage adaptivity in any part of the computational domain, even when resolution transitions reach the free surface. Our methodology ensures symmetry, definiteness and second order accuracy of the discrete Poisson operator, and eliminates numerical and visual artifacts of prior octree schemes. This is achieved by adapting the operators acting on the octree's simulation variables to reflect the structure and connectivity of a power diagram , which recovers primal-dual mesh orthogonality and eliminates problematic T-junction configurations. We show how such operators can be efficiently implemented using a pyramid of sparsely populated uniform grids, enhancing the regularity of operations and facilitating parallelization. A novel scheme is proposed for encoding the topology of the power diagram in the neighborhood of each octree cell, allowing us to locally reconstruct it on the fly via a lookup table, rather than resorting to costly explicit meshing. The pressure Poisson equation is solved via a highly efficient, matrix-free multigrid preconditioner for Conjugate Gradient, adapted to the power diagram discretization. We use another sparsely populated uniform grid for high resolution interface tracking with a narrow band level set representation. Using the recently introduced SPGrid data structure, sparse uniform grids in both the power diagram discretization and our narrow band level set can be compactly stored and efficiently updated via streaming operations. Additionally, we present enhancements to adaptive level set advection, velocity extrapolation, and the fast marching method for redistancing. Our overall framework gracefully accommodates the task of dynamically adapting the octree topology during simulation. We demonstrate end-to-end simulations of complex adaptive flows in irregularly shaped domains, with tens of millions of degrees of freedom.