Harmonic Functions Research Articles

Existence of harmonic maps and eigenvalue optimization in higher dimensions

AbstractWe prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^{n},g)$ ( M n , g ) of dimension $n>2$ n > 2 to any closed, non-aspherical manifold $N$ N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^{k}$ N = S k , $k\geqslant 3$ k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^{k}$ M → S k of index at most $k+1$ k + 1 , with singular set of codimension at least 7 for $k$ k sufficiently large. Furthermore, if $3\leqslant n\leqslant 5$ 3 ⩽ n ⩽ 5 , we show that these smooth harmonic maps stabilize as $k$ k becomes large, and correspond to the solutions of an eigenvalue optimization problem on $M$ M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

Hardy decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Unique continuation problem on RCD Spaces. I

In this note we establish the weak unique continuation theorem for caloric functions on compact RCD(K, 2) spaces and show that there exists an RCD(K, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.

Landau-Bloch type theorem for elliptic and quasiregular harmonic mappings

In this paper, we establish a coefficient bound for quasiregular and elliptic harmonic mappings and using these bounds, we establish Landau-Bloch type theorem for (K,K′)-elliptic and K-quasiregular harmonic mappings in the plane. Furthermore, we prove the coefficient estimates for K-quasiconformal harmonic self maps defined on the unit disc D.

Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

We consider the rigorously derived thin shell membrane \Gamma -limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation m\colon \omega\subset \mathbb{R}^{2}\to \mathbb{R}^{3} and the orthogonal microrotation tensor field R\colon \omega\subset \mathbb{R}^{2}\to \operatorname{SO}(3) . The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet-type energy term |\mathrm{D}R|^{2} . We use Rivière’s regularity techniques of harmonic-map-type systems for our system which couples harmonic maps to \operatorname{SO}(3) with a linear equation for m . The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

If u:Ω⊂Rd→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u: \\Omega \\subset \\mathbb {R}^d \\rightarrow \ extrm{X}$$\\end{document} is a harmonic map valued in a metric space X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{X}$$\\end{document} and E:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}: \ extrm{X}\\rightarrow \\mathbb {R}$$\\end{document} is a convex function, in the sense that it generates an EVI0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{EVI}_0$$\\end{document}-gradient flow, we prove that the pullback E∘u:Ω→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u: \\Omega \\rightarrow \\mathbb {R}$$\\end{document} is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{X}$$\\end{document}, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document} norm of E∘u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u$$\\end{document} on ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} controls the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} norm of E∘u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u$$\\end{document} in 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} for some well-chosen exponents p≥q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\ge q$$\\end{document}, including the case p=q=+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=q=+\\infty $$\\end{document}. In particular, our results apply when E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}$$\\end{document} is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Brenier (Optimal transportation and applications (Martina Franca, 2001), volume 1813 of lecture notes in mathematics, Springer, Berlin, pp 91–121, 2003).

A Classification of Supersymmetric Kaluza–Klein Black Holes with a Single Axial Symmetry

AbstractWe extend the recent classification of five-dimensional, supersymmetric asymptotically flat black holes with only a single axial symmetry to black holes with Kaluza–Klein asymptotics. This includes a similar class of solutions for which the supersymmetric Killing field is generically timelike, and the corresponding base (orbit space of the supersymmetric Killing field) is of multi-centred Gibbons–Hawking type. These solutions are determined by four harmonic functions on $$\mathbb {R}^3$$ R 3 with simple poles at the centres corresponding to connected components of the horizon, and fixed points of the axial symmetry. The allowed horizon topologies are $$S^3$$ S 3 , $$S^2\times S^1$$ S 2 × S 1 , and lens space L(p, 1), and the domain of outer communication may have non-trivial topology with non-contractible 2-cycles. The classification also reveals a novel class of supersymmetric (multi-)black rings for which the supersymmetric Killing field is globally null. These solutions are determined by two harmonic functions on $$\mathbb {R}^3$$ R 3 with simple poles at centres corresponding to horizon components. We determine the subclass of Kaluza–Klein black holes that can be dimensionally reduced to obtain smooth, supersymmetric, four-dimensional multi-black holes. This gives a classification of four-dimensional asymptotically flat supersymmetric multi-black holes first described by Denef et al.

Existence results for the Landau–Lifshitz–Baryakhtar equation

In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.

Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields H1(S,T), where S and T are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface T is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.

Phasor field based photoacoustics: A virtual orthogonal wavefield model for nonnegative photoacoustic computational tomography

The phasor field technique has substantially aided the advancement of the time of flight imaging field and sparked intense scientific research interest during the past several years. Recently, phasor field approach was brought to the field of photoacoustic imaging and results in greatly enhanced image SNR and less artifact interference. This paper aims to establish the foundation for a comprehensive physical framework for phasor field based photoacoustics. To demonstrate the rigorousness of the phasor field based photoacoustics, this work gives a derivation of its virtual forward propagation and backward phasor field reconstruction. Phasor field based photoacoustic system works by creating chunks of initial orthogonal photoacoustic vibration, propagating the virtual orthogonal phasor waves, and performing backward phasor field reconstruction. We present here a formal study of phasor field framework utilizing spherical harmonic function decomposition based point spread function (PSF) analysis and provide a rigorous theoretical explanation for its nonnegative imaging capability of phasor field photoacoustic imaging. Simulation and experimental results reported in the paper support the effectiveness of the phasor field photoacoustic method for nonnegative imaging reconstruction. Moreover, the virtual nature of phasor field method enables flexibility and compatibility with different reconstructive mechanism for optimized imaging performance. Phasor field method the potential to advance the photoacoustic imaging techniques across multiple fields, benefiting various applications and enabling new avenues of research and development.

The range set of zero for harmonic mappings of the unit disk with sectorial boundary normalization

Given a family \(\mathcal F\) of all complex-valued functions in a domain \(\Omega\subset\hat{\mathbb{C}}\), the authors introduce the range set \(RS_{\mathcal F}(A)\) of a set \(A\subset\Omega\) under the class in question, i.e. the set of all \(w\in\Bbb C\) such that \(w\in F(A)\) for a certain \(F\in\mathcal F\). Let \(T_1,T_2,T_3\) be closed arcs contained in the unit circle \(\Bbb T\) of the same length \(2\pi/3\) and covering \(\Bbb T\). The paper deals with the range set \(RS_{\mathcal F}(\{0\})\), where \(\mathcal F\) is the class of all complex-valued harmonic functions \(F\) of the unit disk \(\Bbb D\) into itself satisfying the following sectorial condition: For each \(k\in\{1,2,3\}\) and for almost every \(z\in T_k\) the radial limit \(F^*(z)\) of the function \(F\) at the point \(z\) belongs to the angular sector determined by the convex hull spanned by the origin and arc \(T_k\). In 2014 the authors proved that for any \(F\in\mathcal F\), \(|F(z)|\le\frac{4}{3}-\frac{2}{\pi}\arctan\left(\frac{\sqrt{3}}{1+2|z|}\right), \quad z\in\Bbb D\), by which \(|F(0)|\le 2/3\). This implies that \(RS_{\mathcal F}(\{0\})\) is a subset of the closed disk of radius 2/3 and centred at the origin. In the paper the range set \(RS_{\mathcal F}(\{0\})\) is precisely determined.

On subelliptic harmonic maps with potential

On subelliptic harmonic maps with potential

The wind farm pressure field

Abstract. The disturbed atmospheric pressure near a wind farm arises from the turbine drag forces in combination with vertical confinement associated with atmospheric stability. These pressure gradients slow the wind upstream, deflect the air laterally, weaken the flow deceleration over the farm, and modify the farm wake recovery. Here, we describe the airflow and pressure disturbance near a wind farm under typical stability conditions and, alternatively, with the simplifying assumption of a rigid lid. The rigid lid case clarifies the cause of the pressure disturbance and its close relationship to wind farm drag. The key to understanding the rigid lid model is the proof that the pressure field p(x,y) is a harmonic function almost everywhere. It follows that the maximum and minimum pressure occur at the front and back edge of the farm. Over the farm, the favorable pressure gradient is constant and significantly offsets the turbine drag. Upwind and downwind of the farm, the pressure field is a dipole given by p(x,y)≈Axr-2, where the coefficient A is proportional to the total farm drag. Two derivations of this law are given. Field measurements of pressure can be used to find the coefficient A and thus to estimate total farm drag.

Non-Strict Plurisubharmonicity of Energy on Teichmüller Space

Abstract For an irreducible representation $\rho :\pi _{1}(\Sigma _{g})\to \textrm{GL}(n,\mathbb{C})$, there is an energy functional $\textrm{E}_{\rho }: {{\mathcal{T}}}_{g}\to \mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\textrm{GL}(n,\mathbb{C})/\textrm{U}(n)$. It follows from a result of Toledo that $\textrm{E}_{\rho }$ is plurisubharmonic, that is, its Levi form is positive semi-definite. We describe the kernel of this Levi form, and relate it to the $\mathbb{C}^{*}$ action on the moduli space of Higgs bundles. We also show that the points in $ {{\mathcal{T}}}_{g}$ where strict plurisubharmonicity fails (i.e., this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,\rho )$, the map $\textrm{E}_{\rho }$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

Polyanalytic Neumann-n problem in the unit disk

An iterated Neumann boundary value problem for the polyanalytic operator is investigated in the unit disc of the complex plane. Although this problem can be treated for domains in the complex plane having a harmonic Green function, the unit disk is particularly interesting as it is the origin of complex boundary value problems and the prototype of simply connected domains. Moreover, the resulting formulas for solutions and for solvability conditions are expressed by integral formulas with explicit kernels. By the way, the result verifies the findings for domains with harmonic Green function.

Landau's theorem for harmonic Bloch functions on simply connected domains

For α ∈ ( 0 , ∞ ) , let B H , Ω ( α ) denote the class of harmonic α-Bloch mappings on a proper simply connected domain Ω ⊆ C . In this article, we study coefficient estimates for harmonic α-Bloch mappings on a proper simply connected domain containing the unit disk. Furthermore, we establish the Landau's theorem and coefficient estimates for the harmonic α-Bloch space B H , Ω γ ( α ) on the shifted disk Ω γ , where Ω γ := { z ∈ C : | z + γ 1 − γ | < 1 1 − γ } and 0 ≤ γ < 1 .

Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.

Three-dimensional quasi-static general solution for isotropic chemoelastic materials and their application

Based on potential theory, the three-dimensional quasi-static general solution for isotropic chemoelastic materials is presented in this work. Through the three-dimensional general solution, the Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived. This can serve as theoretical guidance for future engineering practices. Four functions constitute the expressions of the general solution that satisfy the harmonic functions and the quasi-static transport equation, respectively. The Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived by combining the general solution with the chemical balance boundary conditions at infinity. It can be expressed in terms of the error function and elementary functions. Finally, the numerical results are provided, as shown in the contours. These results can be used to analyze the variation law in the coupling fields of isotropic chemoelastic materials. The corresponding analysis can provide a theoretical basis for elucidating the mechanism of the chemoelastic coupling problem in further work.

Bloch-type theorems for meromorphic harmonic mappings

In this paper, we first derive a Landau-type theorem and a Bloch-type theorem for the class of meromorphic harmonic mappings. Then we establish two new versions of Bloch-type theorems for certain K-quasiregular meromorphic harmonic mappings in the unit disk.

Principal Preferred Orientation Evaluation of Steel Materials Using Time-of-Flight Neutron Diffraction

Comprehensive information on in situ microstructural and crystallographic changes during the preparation/manufacturing processes of various materials is highly necessary to precisely control the microstructural morphology and the preferred orientation (or texture) characteristics for achieving an excellent strength–ductility–toughness balance in advanced engineering materials. In this study, in situ isothermal annealing experiments with cold-rolled 17Ni-0.2C (mass%) martensitic steel sheets were carried out by using the TAKUMI and ENGIN-X time-of-flight neutron diffractometers. The inverse pole figures based on full-profile refinement were extracted to roughly evaluate the preferred orientation features along three principal sample directions of the investigated steel sheets, using the General Structure Analysis System (GSAS) software with built-in generalized spherical harmonic functions. The consistent rolling direction (RD) inverse pole figures from TAKUMI and ENGIN-X confirmed that the time-of-flight neutron diffraction has high repeatability and statistical reliability, revealing that the principal preferred orientation evaluation of steel materials can be realized through 90° TD ➜ ND (transverse direction ➜ normal direction) rotation of the investigated specimen on the sample stage during two neutron diffraction experiments. Moreover, these RD, TD, and ND inverse pole figures before and after the in situ experiments were compared with the corresponding inverse pole figures recalculated from the MUSASI-L complete pole figure measurement and the HIPPO in situ microstructure evaluation, respectively. The similar orientation distribution characteristics suggested that the principal preferred orientation evaluation method can be applied to the in situ microstructural evolution of bulk orthorhombic materials and spatially resolved principal preferred orientation mappings of large engineering structure parts.