Properties Of Harmonic Functions Research Articles

Uniqueness of the Solution of the Dirichlet Problem for the Poisson Equation with a Singular -Kipriyanov Operator

Functions satisfying the Laplace equation for the Kipriyanov-operator are said to be-harmonic. The following properties of-harmonic functions are presented and proved: the Green-type integral representation of-functions, the spherical mean theorem, and the maximum principle. As a corollary, the uniqueness of the solution of the interior and exterior Dirichlet problems is proved.

The method of fundamental solutions for analytic functions in complex analysis

<abstract><p>This paper extends the method of fundamental solutions (MFS) for solving the boundary value problems of analytic functions based on Cauchy-Riemann equations and properties of harmonic functions. The conformal mapping technique is applied to introduce the singularities of the approximate analytic functions and reconstruct the fundamental solutions. The presented method can naturally introduce the information of homogeneous boundary conditions and singularity properties, when the conformal mapping technique or the reconstructed fundamental solutions are used. The numerical examples show that the proposed method has the advantages of conciseness, reliability, efficiency, high accuracy and easy-using, respectively. The developed method can be used to solve the boundary value problems (BVPs) of analytic functions without considering single-valuedness, which simplify the numerical analysis.</p></abstract>

Geometric properties of harmonic functions associated with the symmetric conjecture points and exponential function

In this paper, some classes of univalent harmonic functions are introduced by subordination, where the analytic parts of which are exponential starlike (or convex) functions with respect to the symmetric conjecture points. According to the relationships of the analytic part and the co-analytic part, the geometric properties, such as coefficient estimates, distortion theorems, integral expressions, estimates and growth conditions and covering theorem, of the classes are obtained.

Harmonicity and invariance on slices of the Boolean cube

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree harmonic multilinear polynomials (a polynomial in $$x_1,\ldots ,x_n$$ is harmonic if it is annihilated by $$\sum _{i=1}^n \frac{\partial }{\partial x_i}$$ ). Here we provide an alternative proof for general low-degree harmonic multilinear polynomials, with no constraints on the influences. We show that any real-valued harmonic multilinear polynomial on the slice whose degree is $$o(\sqrt{n})$$ has approximately the same distribution under the slice and cube measures. Our proof is based on ideas and results from the representation theory of $$S_n$$ , along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, our decomposition is applied in a non-stationary non-reversible setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the known facts that (1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and (2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is $$o(\sqrt{n})$$ .

Properties of Harmonic Functions Defined by Shear Construction

In this paper, we introduce a new subclass of harmonic functions defined by shear construction. Among other properties of this subclass, we study the convolution of its elements of it with some special subclasses of harmonic functions. Also, we provide coefficient conditions leading to harmonic mappings which are starlike of order \(\alpha \). Some interesting applications of the general results are also presented.

Spline Parameterization of Complex Planar Domains for Isogeometric Analysis

Abstract Isogeometric Analysis (IGA) involves unification of modelling and analysis by adopting the same basis functions (splines), for both. Hence, spline based parametric model is the starting step for IGA. Representing a complex domain, using parametric geometric model is a challenging task. Parameterization problem can be defined as, finding an optimal set of control points of a B-spline model for exact domain modelling. Also, the quality of parameterization, too has significant effect on IGA. Finding the B-spline control points for any given domain, which gives accurate results is still an open issue. In this paper, a new planar B-spline parameterization technique, based on domain mapping method is proposed. First step of the methodology is to map an input (non-convex) domain onto a unit circle (convex) with the use of harmonic functions. The unique properties of harmonic functions: global minima and mean value property, ensures the mapping is bi-jective and with no self-intersections. Next step is to map the unit circle to unit square to make it apt for B-spline modelling. Square domain is re-parameterized by using conventional centripetal method. Once the domain is properly parameterized, the required control points are computed by solving the B-spline tensor product equation. The proposed methodology is validated by applying the developed B-spline model for a static structural analysis of a plate, using isogeometric analysis. Different domains are modelled to show effectiveness of the given technique. It is observed that the proposed method is versatile and computationally efficient.

Lower Estimates for Certain Harmonic Functions in the Half Space

We will give the growth properties of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Levin in a half plane.

An iterative spherical mean value method for background field removal in MRI

The sophisticated harmonic artifact reduction for phase data (SHARP) method has been proposed for the removal of background field in MRI phase data. It relies on the spherical mean value (SMV) property of harmonic functions, and its accuracy depends on the radius of the sphere used for computing the SMV and truncation threshold needed for deconvolution. The goal of this study was to develop an alternative SMV-based background field removal method with reduced dependences on these parameters. The proposed background field removal method, termed iterative SMV (iSMV), consists of applying the SMV operation repeatedly on the field map. It was validated in a phantom and in vivo brain data of five healthy volunteers. The iSMV method demonstrates accurate background field removal in the phantom. Compared with SHARP, the iSMV method shows a significantly reduced dependence on the SMV radius both in phantom and in human data. Because a smaller radius can be chosen, the iSMV method allows retaining a larger part of the region of interest compared with SHARP. The iSMV method is an effective background field removal method with a reduced dependence on method parameters. Magn Reson Med 72:1065-1071, 2014. © 2013 Wiley Periodicals, Inc.

Growth of certain harmonic functions in an n-dimensional cone

We give the growth properties of harmonic functions at infinity in a cone, which generalize the results obtained by Siegel-Talvila.

Hadamard’s formula and couplings of SLEs with free field

The relation between level lines of Gaussian free fields (GFF) and SLE4-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE4 is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.

Properties of harmonic functions which are convex of order $ \bf \beta $ with respect to symmetric points

Let $ \mathcal{H} $ denote the class of functions $ f $ which are harmonic and univalent in the open unit disc $ {D=\{z:|z|<1\}} $. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in $ \mathcal{D} $ and are related to the functions convex of order $ \beta(0\leq \beta <1) $, with respect to symmetric points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

Theory and application of pseudo‐damping in structures

This presentation describes the underlying physics and theory of pseudo damping to reduce vibration of conservative structures. The theoretical approach is based on particular properties of harmonic functions that lead to construction of a complex system that exhibits nearly irreversible energy transfer. This approach also provides the ability to tailor energy absorption from a particular structure in the absence of damping. The authors will also discuss extension of the concept to continuous systems and its application to waveguides. The presentation will illustrate test results including application to reduce vibrations in a satellite. [Research carried out while AA served at NSF.]

On non-commutative extensions of $\widehat{\G}_a$ by $\widehat{\Gg}^{(M)}$\\ over an ${\F}_p$-algebra

Potential theory on a Cartier tree T is developed on the lines of the classical and the axiomatic theories on harmonic spaces. The harmonic classifications of such trees are considered; the notion of a subordinate structure on T is introduced to consider more generally the potential theory on T associated with the Schrodinger equation Duðx Þ¼ QðxÞuðxÞ, QðxÞb 0o nT; polysuperharmonic functions and poly- potentials on T are defined and a Riesz-Martin representation for positive polysuper- harmonic functions is obtained. In this note, we study some classical potential-theoretic concepts like balayage, domination principle etc. in the context of a tree T and introduce the notions of polysuperharmonic functions and polypotentials on T and obtain some of their properties. The tree T is taken in the sense of Cartier's (4), a graph with infinite vertices, connected, locally finite and no circuits, provided with a transition probability structure. Bajunaid et al. (1) show that the har- monic functions on the vertices of T can be linearly extended to the edges, so that the extended functions verify the axioms 1, 2, 3 of Brelot. Consequently, some of the properties of harmonic functions and potentials on T can be im- mediately deduced from the axiomatic potential theory. However, on many occasions, direct proofs of theorems about harmonic functions on T are simpler and give more informations in comparison to those deduced from the axiomatic theory. Secondly, some theorems in the axiomatic theory require more assumptions than the axioms 1, 2, 3 only. One such is the converse to the Riesz representation theorem in a harmonic space W which states that given a positive Radon measure m on an open set o in W, there exists a superharmonic function s on o with associated measure m in a local Riesz representation. To prove this, we need the axiom of analyticity (de La Pradelle (6)) which is not generally valid on T. However, this converse to the Riesz representation is true on T (Theorem 2.4). Thirdly, for polyharmonic

Classical image treatment of a geometry composed of a circular conductor partially merged in a dielectric cylinder and related problems in electrostatics

The classical treatment of the Laplace equation by electrical inversion is used to obtain analytical expressions for the two-dimensional perturbed electrostatic fields in the presence of a two-phase 2D object. The geometry of the two-phase object is composed of two overlapping infinitely long cylinders Ca and Cb of radii a and b, respectively, intersecting at a vertex angle . The composite inclusion has the shape resembling a 2D snowman type of object with a conducting cylinder Ca partly protruded into the dielectric cylinder Cb with a dielectric constant different from that of the host medium. The mathematical problem with this inclusion in the electrostatic environment is formulated in terms of electric potentials (and complex potentials as well) with the mixed Dirichlet and Neumann boundary conditions at the boundary of the hybrid object. General expressions for the perturbed electrostatic fields in the two phases are obtained in a straightforward fashion using Kelvin's inversion together with shift and reflection properties of harmonic functions. The general results are exploited to derive solutions for the hybrid object embedded in (i) a transverse or a longitudinal uniform field and (ii) a field generated by a dipole. The dipole coefficients are extracted directly from the exact solutions in a fairly simple manner. The plots of the normalized polarizabilities (derived from dipole coefficients) for the imposed uniform field show that the polarizability in the longitudinal direction is greater or less than the transverse polarizability depending on the dielectric constants of the composite body and the host medium. The equipotential plots generated using the exact solutions also show fascinating field patterns. An interesting connection between the present solutions and the cyclic groups in abstract algebra is outlined as well in the appendix.

Properties of the mappings that are close to the harmonic mappings

We continue studying the mappings that are close to the harmonic mappings (e-quasiharmonic mappings with e small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of e-quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for e-quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of e-quasiharmonic mappings with e small which preserves proximity to the harmonic mappings.

Global Representation of Harmonic and Biharmonic Functions

Using some well-known properties of harmonic functions, such as the maximum principle and the existence of the Dirichlet solution, certain representation theorems for harmonic and biharmonic functions are proved in \(\mathbb{R}^n \).

Martin compactification for discrete potential theory and the mean value property

A converse of the well-known theorem on themean value property of harmonic functions is given. It is shown that a positive measurable function is harmonic if it possesses arestricted mean value property. Earlier proofs obtained using the probabilistic techniques were given by Veech, Heath and Baxter. Our approach is based on a Martin type compactification built up with the help of some quite elementarya priori inequalities foraveraging kernels.

On a Property of Harmonic Functions

If we divide the space \mathbb R^n into two disjoint areas with one common hypersurface and define a harmonic function in each part of these areas such that their gradients vanish at infinity and the normal components of their gradients are equal on the hypersurface, then for some hypersurfaces such as a circle in \mathbb R^2 or a hyperplane in \mathbb R^n the sum of the tangential components of the gradients is zero. We investigate for which hypersurfaces we have this property and prove that such hypersurfaces in \mathbb R^2 are only circles and straight lines. We also give an application of this property to an ideal plane flow through a porous surface.

A harmonic quadrature formula characterizing open strips

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, thenA converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such thatfor every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

A field problem with a non‐linear boundary condition

A non‐linear field problem is posed, in which the non‐linearity occurs in the boundary condition imposed. Because the non‐linearity is expressed in terms of a harmonic function of the components of the field, it is possible to obtain an analytical solution using elementary properties of harmonic functions. The problem and its solution furnish a suitable teaching example for engineers and physicists.