Classical Potential Theory Research Articles

On an extension of Fuglede’s theory of pseudo-balayage and its applications

For suitable function kernels κ on a locally compact space, we develop a theory of inner pseudo-balayage ω̂A of signed (Radon) measures ω of finite energy onto a quasiclosed set A, ω̂A being defined as the solution to the problem of minimizing the Gauss functional ∫κ(x,y)d(μ⊗μ)(x,y)−2∫κ(x,y)d(μ⊗ω)(x,y),where μ ranges over all positive measures of finite energy concentrated on the set A. If A is Borel, the concept of inner pseudo-balayage ω̂A is shown to coincide with that of outer pseudo-balayage, introduced in Fuglede’s work (Fuglede, 2016), which was however only concerned with ω⩾0, whereas the investigation of signed ω requires essentially different methods and approaches. The theory of pseudo-balayage thereby established enables us to improve substantially our recent results on the well-known inner Gauss variational problem (Zorii, 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications.

Towards a momentum potential theory for reacting flows

Mutual coupling of (thermofluiddynamic) modes of perturbations can affect the thermo-acoustic stability of combustors and contribute to combustion noise. For example, vortical or entropic perturbations can be transferred to acoustic perturbations if accelerated by the mean flow. The decomposition of perturbation fields into the respective modes and a linear description of their interactions in terms of fluctuating primitive variables is challenging. In contrast, Doak’s momentum potential theory promises an unambiguous decomposition in terms of momentum fluctuations, which is not limited to the linear regime. Whereas classical momentum potential theory takes into account hydrodynamic, acoustic and entropic modes in unconfined flows, the investigation of noise generation in combustion chambers requires the extension of the momentum potential theory to capture modes linked to the fluctuation of species mass fractions (“species mode”) arising from the change in chemical composition due to the reaction. Furthermore, a rigorous treatment of boundary conditions due to the confinement of the flow inside the combustor is required. The herein presented extension to reactive flows consists of two steps, (i) the formulation of a potential for momentum fluctuations related to species modes and (ii) identification of the total fluctuating enthalpy related to species modes. The extended theory is applied to post-process computational fluid dynamic simulation data of the propagation of entropy and species perturbations through one-dimensional ducts, nozzles and premixed flames. We find that although momentum potential theory offers a complete decomposition of momentum perturbations for reactive flows, the meaningful interpretation of this decomposition is rather challenging, even for non-reactive flows.

Harmonic models and molecular dynamics simulations of isomorph behavior of Lennard-Jones fluids: Excess entropy and high temperature limiting behavior

Henchman’s approximate harmonic model of liquids is extended to predict the thermodynamic behavior along lines of constant excess entropy (“isomorphs”) in the liquid and supercritical fluid regimes of the Lennard-Jones (LJ) potential phase diagram. Simple analytic expressions based on harmonic cell models of fluids are derived for the isomorph lines, one accurate version of which only requires as input parameters the average repulsive and attractive parts of the potential energy per particle at a single reference state point on the isomorph. The new harmonic cell routes for generating the isomorph lines are compared with those predicted by the literature molecular dynamics (MD) methods, the small step MD method giving typically the best agreement over a wide density and temperature range. Four routes to calculate the excess entropy in the MD simulations are compared, which includes employing Henchman’s formulation, Widom’s particle insertion method, thermodynamic integration, and parameterized LJ equations of state. The thermodynamic integration method proves to be the most computationally efficient. The excess entropy is resolved into contributions from the repulsive and attractive parts of the potential. The repulsive and attractive components of the potential energy, excess Helmholtz free energy, and excess entropy along a fluid isomorph are predicted to vary as ∼T−1/2 in the high temperature limit by an extension of classical inverse power potential perturbation theory statistical mechanics, trends that are confirmed by the MD simulations.

PRESENTATION OF THE GRAVITY FIELD OF CELESTIAL BODIES USING THE POTENTIALS OF FLAT ELLIPSOIDAL DISCS

One of the possible ways for representing the external gravitational field of the planet by the potentials of flat discs, based on the classical potential theory, is proposed. At the same time, the potentials of a single- and double-layer are used for the description with the placement of the integration regions in the equatorial plane. The coefficients of the series expansion of these functions are linear combinations of the Stokes constants of the gravitational field and are uniquely expressed in terms of them. Series terms are single- or double-layer potentials. This makes it possible to calculate these terms using the results of the ellipsoid potential theory. The convergence of such a series, in contrast to the traditional one for spherical functions, is much wider and practically covers the effect of the external potential excluding the region of integration, including in the superficial parts of the surface. Since there is no problem with the convergence of the obtained expansions, we can interpret the obtained results more fully. The construction of flat density distributions for the potentials of a single and double layer is an additional tool in the study of the internal structure of the celestial body, as it is essentially a projection of the volume density of the planet’s interior onto the equatorial plane. Therefore, the extrema of these functions combine the features of the three-dimensional distribution function of the planet’s interior

Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities

The main aim of this paper is to study the geometric and metric properties of $B$- and $C$-capacities related to problems of uniform approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of Euclidean spaces. In the harmonic case this problem is well known, and it was studied in detail in the framework of classical potential theory in the first half of the 20th century. For a wide class of equations mentioned above, we obtain two-sided estimates between the corresponding $B_+$- and $C_+$-capacities (defined in terms of potentials of positive measures) and the harmonic capacity in the same dimension. Our research method is based on new simple explicit formulae obtained for the fundamental solutions of the equations under consideration. Bibliography: 12 titles.

Steady Two-Dimensional Conduction: Simple and Double Layer Potentials, Corner Singularities, and Induced Heat Flux

Abstract Two-dimensional steady-state heat conduction is possible outside closed boundaries on which two isothermal segments at different temperatures are separated by two adiabatic segments. Remarkably, previous research showed that the conduction shape factor for the region exterior to the boundary is equal to that for the interior region, despite the asymmetry and singularity of the boundary heat flux distributions. In this study, classical potential theory is used for the temperature and heat flux distributions as a combination of simple-layer and double-layer potentials, including the relationships between the values inside and outside the boundary curve. Isothermal boundaries exhibit an induced heat flux that varies from point-to-point on the boundary. The induced flux integrates to zero over each isothermal edge. Singularities of the heat flux are identified and resolved. Computations that validate the theory are provided for mixed boundary conditions on a disk and a square. Numerical fits to both the simple-layer and double-layer densities are given for the disk and the square. The analysis explains why the interior and exterior conduction shape factors are equal despite wildly differing heat flux distributions, and the results are compared to a previous study of this configuration. This paper also develops fundamental concepts of potential theory and can serve as a tutorial on the subject.

On the theory of balayage on locally compact spaces

The paper deals with the theory of balayage of (signed) Radon measures μ of finite energy on a locally compact space X with respect to a consistent kernel κ satisfying the domination principle. Such theory is now specified for the case where the topology on X has a countable base, while any f ∈ C0(X), a continuous function on X of compact support, can be approximated in the inductive limit topology on the space C0(X) by potentials $\kappa \lambda :=\int \limits \kappa (\cdot ,y) d\lambda (y)$ of measures λ of finite energy. In particular, we show that then the inner balayage can always be reduced to balayage to Borel sets. In more details, for arbitrary A ⊂ X, there exists a Kσ-set A0 ⊂ A such that $$ \mu^{A}=\mu^{A_{0}}=\mu^{*A_{0}}\ \text{for all } \mu, $$ μA and μ∗A denoting the inner and the outer balayage of μ to A, respectively. Furthermore, μA is now uniquely determined by the symmetry relation $\int \limits \kappa \mu ^{A} d\lambda =\int \limits \kappa \lambda ^{A} d\mu $ , λ ranging over a certain countable family of measures depending on X and κ only. As an application of these theorems, we analyze the convergence of inner and outer swept measures and their potentials for monotone families of sets. The results obtained do hold for many interesting kernels in classical and modern potential theory on $\mathbb {R}^{n}$ , $n\geqslant 2$ .

Electronic structures and thermoelectric properties of heavily doped n-type ZrCoBi from first principles calculations

We present a theoretical investigation on the electronic structures and the thermoelectric properties of doped ZrCoBi half-Heusler compound using a combination of density functional theory calculations, semi classical Boltzmann transport theory and deformation potential theory. The effect of doping is described by a combination of rigid band approximation (RBA) and a model that explicitly includes the dopant atom within the lattice of ZrCoBi. The results demonstrate a high power factor with considerable n- and p-doping concentrations at high temperatures. Our findings further reveal that the electronic transport coefficients of ZrCo0.75M0.25Bi (M = Ni, Pd, Pt) are comparable because of their identical electronic structures. At 900 K, a high figure of merit (ZT) of ~ 0.35, ~ 0.38, and ~ 0.40 are obtained for ZrCo0.75Ni0.25Bi, ZrCo0.75Pd0.25Bi and ZrCo0.75Pt0.25Bi respectively.

Continuous operator method application for direct and inverse scattering problems

Abstract. We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach spaces, including in cases when the Frechet (Gateaux) derivative of a nonlinear operator is irreversible in a neighborhood of the initial value. In this paper, it is applied to the solution of the Dirichlet and Neumann problems for the Helmholtz equation and to determine the wave number in the inverse problem. The internal and external problems of Dirichlet and Neumann are considered. The Helmholtz equation is considered in domains with smooth and piecewise smooth boundaries. In the case when the Helmholtz equation is considered in domains with smooth boundaries, the existence and uniqueness of the solution follows from the classical potential theory. When solving the Helmholtz equation in domains with piecewise smooth boundaries, the Wiener regularization is carried out. The Dirichlet and Neumann problems for the Helmholtz equation are transformed by methods of potential theory into singular integral equations of the second kind and hypersingular integral equations of the first kind. For an approximate solution of singular and hypersingular integral equations, computational schemes of collocation and mechanical quadrature methods are constructed and substantiated. The features of the continuous method are illustrated with solving boundary problems for the Helmholtz equation. Approximate methods for reconstructing the wave number in the Helmholtz equation are considered.

Boundary integrals for oscillating bodies in stratified fluids

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

A Low Discrepancy Sequence on Graphs

Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.

Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances

The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.

Nonlinear $n$-term approximation of harmonic functions from shifts of the Newtonian kernel

A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian kernel with poles outside $\overline{B^d}$ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.

Developments and perspectives in Nonlinear Potential Theory

Nonlinear Potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered. In recent years there has been a substantial development of this subject, mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials. Here we give a brief account of such developments and outline the connections with different fields.

Determination of Redox Currents in Electroless Systems: Correct Application of Mixed Potential Analysis

Electroless processes have traditionally been modeled by way of the ‘mixed potential theory’, stipulating that the electroless process operates at a potential where the anodic and cathodic currents are of equal magnitude. The latter are typically measured in separate systems, with compositions similar to the process chemistry, however, absent either the oxidized or the reduced species. While valid for some processes, this application of the mixed potential theory fails in many others. A new method for characterizing electroless processes is presented here. Applying external current to the complete electroless chemistry, both the oxidation and reduction processes proceed in parallel, via the external current and through the electroless process. Monitoring the external current and the amount of deposited metal, the correct anodic and cathodic polarization curves in the actual system are determined and applied to the mixed potential theory. Using this method, excellent agreement was found in characterizing the process of electroless copper reduction by glyoxylic acid, a system which the classical mixed potential theory fails to correlate. Although applied here just to the copper - glyoxylic acid system, this technique for generating polarization curves in complete electroless systems is general and can be extended to other electroless and redox processes.

Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space–time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires $$O(N_S^2N_T^2)$$ work and $$O(N_SN_T)$$ storage, where $$N_S$$ is the total number of discretization points on the spatial boundary and $$N_T$$ is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts—the local part containing the temporal integration from $$t-\delta $$ to t and the history part containing the temporal integration from 0 to $$t-\delta $$ . The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples.

The influence of iodate ion additions to the bath on the deposition of electroless nickel on mild steel

ABSTRACTAn alkaline hypophosphite bath (0.1 M nickel sulphate, 0.2 M sodium hypophosphite, 0.2 M sodium acetate and 0.1 M malic acid, adjusted to pH 5) was used to produce Ni–P coatings on uncoated and electroless nickel pre-plated mild steel. The deposition was monitored by open-circuit potential-time monitoring vs. a saturated calomel reference electrode and potentiostatic current–time monitoring together with anodic and cathodic polarisation. Classical mixed potential theory was applied to the polarisation data to calculate the effect of controlled iodate ion additions (0–1000 ppm) as an accelerator to the electrolyte on the plating rate. The mixed potential and deposition current density increased gradually with potassium iodate concentration. The use of electrochemical data allowed the optimum iodate additive concentration to be established using simple instrumentation.

Quantitative stability of the free boundary in the obstacle problem

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $\mathbb{R}^n$ ($n \ge 2$) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field. To do so, working in the setting of the whole space, we examine the evolution of the free boundary $\Gamma^t$ corresponding to the boundary of the contact set for a family of obstacle functions $h^t$. Assuming that $h=h^t (x) = h(t,x)$ is $C^{k+1,\alpha}$ in $[-1,1]\times \mathbb{R}^n$ and that the initial free boundary $\Gamma^0$ is regular, we prove that $\Gamma^t$ is twice differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that the "normal velocity" and the "normal acceleration" of $\Gamma^t$ are respectively $C^{k-1,\alpha}$ and $C^{k-2,\alpha}$ scalar fields on $\Gamma^t$. This is accomplished by deriving equations for these velocity and acceleration and studying the regularity of their solutions via single and double layers estimates from potential theory.

Semipolar Sets and Intrinsic Hausdorff Measure

Given a “Green function” G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that $G\nu :=\int G(\cdot ,y)\,d\nu (y)$ is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with $m_{G}(A)<\infty $ is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.

Dynamic assessment of nonlinear typical section aeroviscoelastic systems using fractional derivative-based viscoelastic model

Nonlinear aeroelastic systems are prone to the appearance of limit cycle oscillations, bifurcations, and chaos. Such problems are of increasing concern in aircraft design since there is the need to control nonlinear instabilities and improve safety margins, at the same time as aircraft are subjected to increasingly critical operational conditions. On the other hand, in spite of the fact that viscoelastic materials have already been successfully used for the attenuation of undesired vibrations in several types of mechanical systems, a small number of research works have addressed the feasibility of exploring the viscoelastic effect to improve the behavior of nonlinear aeroelastic systems. In this context, the objective of this work is to assess the influence of viscoelastic materials on the aeroelastic features of a three-degrees-of-freedom typical section with hardening structural nonlinearities. The equations of motion are derived accounting for the presence of viscoelastic materials introduced in the resilient elements associated to each degree-of-freedom. A constitutive law based on fractional derivatives is adopted, which allows the modeling of temperature-dependent viscoelastic behavior in time and frequency domains. The unsteady aerodynamic loading is calculated based on the classical linear potential theory for arbitrary airfoil motion. The aeroelastic behavior is investigated through time domain simulations, and subsequent frequency transformations, from which bifurcations are identified from diagrams of limit cycle oscillations amplitudes versus airspeed. The influence of the viscoelastic effect on the aeroelastic behavior, for different values of temperature, is also investigated. The numerical simulations show that viscoelastic damping can increase the flutter speed and reduce the amplitudes of limit cycle oscillations. These results prove the potential that viscoelastic materials have to increase aircraft components safety margins regarding aeroelastic stability.