Harmonic functions on compact sets

In this article we consider classes of harmonic and subharmonic functions introduced with using integral operators Riman-Liouville by Professor M. Djrbashyan when α > 0. These classes are significant generalizations of already well known classes of harmonic and subharmonic functions match up with them only in a particular case. In our article we consider angular and chordal limits of harmonic and subharmonic functions got by using Riman-Liouville integral operators. A set of the points at which, probably, these limits don’t exist are characterized by using a linear measure of zero.

This survey paper deals with the unitary treatment of some classes of linear partial differential equations on Klein surfaces, which are the most general two-manifolds that support harmonic functions. We are mainly concerned with the study of harmonic functions with Dirichlet or Neumann boundary conditions. In such a way, the present paper extends several classical results to the abstract setting of dianalytic manifolds of complex dimension 1. We bring together systematically and concisely the Green and Neumann functions, the harmonic kernel function, and the harmonic measure on Klein surfaces. The technique to extend these concepts is to apply the classical methods to the complex double of a Klein surface, which is a Riemann surface endowed with an antianalytic involution. The existence of a harmonic function on a Riemann surface is given by the property that a harmonic function remains invariant under biholomorphic mappings. Similar problems on Riemann surfaces were developed by L.V. Ahlfors and L. Sario. The analysis developed in this paper offers perspectives to the qualitative analysis of other classes of linear or nonlinear elliptic equations on Klein surfaces.

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

Potential distributions are found for steady wells idealized as sinks, and tapping uniform confined aquifers into which there is leakage in proportion to the drawdown. Several plane regions are treated, along whose boundaries the potential, flux, or both are specified. The ratio of the conductivity of the aquifer to that of the confining bed is assumed very large so that the refraction of flow lines may be assumed complete as they cross the contact. The leakage, thus considered vertical through the confining bed, is assumedly distributed throughout the thickness of the aquifer as it thus aguments or diminishes the horizontal flow, which is then treated as two dimensional.Starting with the fundamental solution for an infinite plane region, the solutions for other plane regions are built up by the method of images. The Green's functions for leaky systems are thus obtained for infinite half‐planes (two cases), infinite quadrants (three cases), infinite strips (three cases), infinite half‐strips (six cases). and rectangles (six cases). Several graphs are given showing potential distributions in special cases and showing the magnitude of drawdown at the well face for various values of the parameters.The result obtained in Part I of this paper for the infinite plane is extended by the method of images to find solutions for infinite strips, infinite half‐strips and for rectangles, in Part II. These solutions are put in forms of series which are convenient for computations. These series are uniformly convergent for all values of x and y except when both are simultaneously taken at the center of the well in which case the series diverges logarithmically to negative infinity. Each of these solutions satisfies the properties of one of the three different types of Green's functions—the first, the second, or the mixed type, but instead of being a harmonic function it satisfies the equation of motion of ground water with linear leakage. Accordingly these solutions may be considered as the Green's functions for systems with linear leakage.

Green's functions play an important role in electroelastic analyses of pyroelecric media. However, most works available on this topic are on case of identical temperature. Based on the 2D and 3D general solution of pyroelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the 2D and 3D Green's function for a steady line heat source and a steady point heat source in semi-infinite, infinite and two-phase pyroelectric plane and media is presented by newly induced harmonic functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the 2D and 3D general solution, and the undetermined constants can be obtained by boundary conditions, continuous conditions and equilibrium conditions.

Green's functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on this topic are on the case of identical temperature. Based on the compact general solution of transversely isotropic piezothermoelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional Green's function for a steady point heat source in a two-phase piezothermoelectric media is presented by eight newly induced harmonic functions. Numerical results are given graphically by contours.

A minimization problem for the Ginzburg-Landau functional is considered on an annulus with thickness R in the class of functions \( \mathcal{F}^d\) where only the degree (the winding number), d, on the boundary is prescribed. The existence of a critical thickness R c which depends on the Ginzburg-Landau parameter δ and d is established. It is shown that if R>Rc then any function in separated form in polar coordinates can not be a global minimizer in the class \( \mathcal{F}^d\) and therefore the functions in a minimizing sequence can not be of this form. Hence for a sufficiently thick annulus, the variational Ginzburg-Landau problem is not a regular perturbation, in δ, of the corresponding problem for harmonic maps. The proof is based on an explicit construction of a sequence of admissible functions with vortices approaching the boundary. The key point of the proof is that the Ginzburg-Landau energy of the functions in this sequence approaches a constant independent of R and δ. By contrast, in the class of functions with separated form in polar coordinates and the same boundary conditions, there exists a unique minimizer whose energy depends on R. This minimizer is rotationally invariant and converges uniformly to the solution of the corresponding harmonic map problem as δ→0.KeywordsCritical ThicknessBlaschke ProductUnique MinimizerSeparate FormRegular PerturbationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions.

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

We consider the computation of the harmonic 3D spatial-domain Green's function of the surface of a piezoelectric substrate. Starting from the Green's function expressed in the spectral domain, the singular contributions are isolated and treated separately. It is found that the surface acoustic wave contributions give rise to an anisotropic generalization of the Hankel function H (2) 0 , the spatial Green's function for the scalar two-dimensional wave equation. The asymptotic behavior at infinity and at the origin are also explicitely treated. The remaining non-singular part of the spatial Green's function is obtained numerically by a combination of fast Fourier transform and quadrature. Illustrations are given in the case of a substrate of Y + 128-cut lithium niobate.

How is it possible to describe any analytic or harmonic function on a compact set in terms of much simpler “construction blocks” like polynomials? Or, how is it possible to know the values of a function inside a compact domain, by knowing only its values on the boundary? Well, these simplifications are possible because the “bricks” are actually organized in complicated and versatile structures. For example the B ∗ algebras. And in addition, the compact domains are certainly among the simples ones, being always reducible to finite reunions. Actually, a complicated structure like a B ∗ algebra, defined by 24 axioms (out of which 13 axioms on commutative algebras, five axioms on norm, one for completeness, and five more specific axioms) can be realized by continuous functions defined on a compact set. It is not the only example. The space l 1 of complex sequences with norm given by the sum of the modules of the terms is isomorphic with the algebra of functions whose Fourier series is absolutely convergent. Also, a compact Hausdorff space, with topology induced by distance, is homeomorphic with a compact subset of [0, 1]N. Any two separable Hilbert spaces are isomorphic, and so on. These similarities bring a unifying point of view: objects of apparently distinct nature, like Weierstrass–Stone theorem on function approximation, Wiener theorem on absolutely convergent Fourier series, spectral expansion of self-adjoint operators, the theorems of Tikhonov, Stone–Čech, or the fixed-point theorem of Brouwer, the Cauchy formula on complex functions, the Green representation theorem, the Poincaré Lemma, etc. actually provide the same fundamental truth: simplification by approximation is possible on compacts.KeywordsHarmonic FunctionCotangent BundleRepresentation FormulaExterior DerivativeCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a harmonic function U in a domain Ω in Euclidean space, the problem of finding a harmonic conjugate V, generalizing the well-known case of the complex plane, was considered in [4] in the framework of Clifford analysis. By the nature of the given construction, which is genuinely cartesian, this approach lead to geometric constraints on the domain Ω. In this paper we consider the problem in a larger class of domains, by a spherical approach. Starting from a real-valued function u, and singling out the radial direction, we explicitly construct a harmonic function of the form w = erv, with v ∈ span(eθ1,…, eθm−1), such that u+w is monogenic, i.e. a null solution of the Dirac operator. As an illustration, the construction is applied to important classes of homogeneous monogenic polynomials and functions. Finally, it is investigated to which extent the approach also applies to the complex plane case.

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses m m values of linear functionals (integrals over hyperspheres) and is exact for all 2 m 2m -harmonic functions, and consequently, for all algebraic polynomials of n n variables of degree 4 m − 1 4m-1 .

In this article we investigate some classes of meromorphic or complex harmonic functions with a pole, which are generated either by analytic conditions or by “coefficient inequalities”. There are given theorems, which combine the geometrical properties of functions of the introduced classes. Some results broaden knowledge about the classes of functions, which were investigated in [

In this paper, we consider classes of harmonic convex functions and give their special characterizations. Furthermore, we consider Hermite Hadamard type inequalities related to these classes to give some non-numeric estimates of well-known definite integrals.