Problem For Harmonic Functions Research Articles

Singularity formation in the harmonic map flow with free boundary

abstract: In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary $$ (0.1)\hskip77pt\cases{u_t=\Delta u& in $\Bbb{R}^2_+\times (0,T)$,\cr u(x,0,t)\in\Bbb{S}^1& for all $(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$,\cr {du\over dy}(x,0,t)\perp T_{u(x,0,t)}\Bbb{S}^1& for all $(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$,\cr u(\cdot, 0)=u_0& in $\Bbb{R}^2_+$} $$ for a function $u:\Bbb{R}^2_+\times [0,T)\to\Bbb{R}^2$. Here $u_0 :\Bbb{R}^2_+\to\Bbb{R}^2$ is a given smooth map and $\perp$ stands for orthogonality. We prove the existence of initial data $u_0$ such that (0.1) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Chen and Lin.

Beltrami equation for the harmonic diffeomorphisms between surfaces

We study harmonic maps between surfaces, that are solutions to a nonlinear elliptic PDE. In Refs. Minsky (1992); Wolf (1989) it was proved that harmonic diffeomorphisms, with nonvanishing Hopf differential, satisfy a Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami coefficient coincides with the imaginary part of the logarithm of the Hopf differential. Therefore, it is a harmonic function. The real part of the logarithm of the Beltrami coefficient satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is the elliptic sinh-Gordon equation.In this paper we also prove the converse: if the imaginary part of the logarithm of the Beltrami coefficient is a harmonic function, then the target surface can be equipped with a metric, conformal to the original one, and the solution of the Beltrami equation is a harmonic map. Therefore, solving a certain Beltrami equation is equivalent to solving the harmonic map problem. Harmonic maps to a constant curvature surface are therefore classified by the classification of the solutions of the elliptic sinh-Gordon equation.The general problem of solving the sinh-Gordon equation is a nonlinear problem and is still open. Different well-known harmonic maps to the hyperbolic plane are proved to be related to the one-soliton solutions of the elliptic sinh-Gordon equation. Moreover, an example is proposed which does not belong to the one-soliton solution of the elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way, for positive, negative and zero curvature of the target surface.

Eigenfunctions of the Laplace Operator and Harmonic Functions on Model Riemannian Manifolds

This article explores and develops opportunities Fourier method of separation of variables for the study of the asymptotic behavior of harmonic functions on noncompact Riemannian manifolds of a special form. These manifolds generalize spherically symmetric manifold and are called model ones in a series of works. In the first part of the paper, an estimate of the eigenfunctions of the Laplace operator is obtained on compact Riemannian manifolds $$S$$ in the norm $$C^{m}(S)$$ . In the second part of the paper, the conditions for the unique solvability of the Dirichlet problem for harmonic functions on model manifolds with smooth boundary data at ‘‘infinity’’ are found. It was shown that the solution of this boundary value problem converges to the boundary data in the $$C^{1}$$ -norm.

Rigidity theorem for harmonic maps with complex normal boundary conditions

Let [Formula: see text] be the open unit ball in [Formula: see text] and let [Formula: see text] be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map [Formula: see text] satisfying the boundary condition that [Formula: see text] on [Formula: see text]. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on [Formula: see text].

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.

Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &\text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$% Here $\varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1})$ for a fixed $1< p< \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a< |x|< b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1}$ with $\rom{Q}$ some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.

Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in five dimensions

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space |$L(p,q)$| topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically |$S^1\times S^2$|⁠. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: |$S^3$|⁠, |$S^1\times S^2$|⁠, or |$L(p,q)$|⁠. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space |$SL(3,\mathbb{R})/SO(3)$|⁠. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.

Compactness results for static and dynamic chiral skyrmions near the conformal limit

We examine lower order perturbations of the harmonic map problem from $$\mathbb {R}^2$$ to $$\mathbb {S}^2$$ including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizers, and their asymptotic limit. Finally we examine dynamic stability and compactness of almost minimizers in the context of the Landau–Lifshitz–Gilbert equation including spin-transfer torques arising from the interaction with an external current.

On the convexity and feasibility of the bounded distortion harmonic mapping problem

Computation of mappings is a central building block in many geometry processing and graphics applications. The pursuit to compute mappings that are injective and have a controllable amount of conformal and isometric distortion is a long endeavor which has received significant attention by the scientific community in recent years. The difficulty of the problem stems from the fact that the space of bounded distortion mappings is nonconvex. In this paper, we consider the special case of harmonic mappings which have been used extensively in many graphics applications. We show that, somewhat surprisingly, the space of locally injective planar harmonic mappings with bounded conformal and isometric distortion has a convex characterization. We describe several projection operators that, given an arbitrary input mapping, are guaranteed to output a bounded distortion locally injective harmonic mapping that is closest to the input mapping in some special sense. In contrast to alternative approaches, the optimization problems that correspond to our projection operators are shown to be always feasible for any choice of distortion bounds. We use the boundary element method (BEM) to discretize the space of planar harmonic mappings and demonstrate the effectiveness of our approach through the application of planar shape deformation.

A solution to Sheil-Small’s harmonic mapping problem for polygons

The problem of mapping the interior of a Jordan polygon univalently by the Poisson integral of a step function was posed by T. Sheil-Small (1989). We describe a simple solution using “ear clipping” from computational geometry.

Regularity and long-time behavior of nonlocal heat flows

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow approach to an optimal eigenvalue partition problem that, in light of the work of Caffarelli and Lin, is equivalent to a constrained harmonic mapping problem. In particular, we show that weak solutions of our heat equation converge as time approaches infinity to stationary solutions of this mapping problem. For these weak solutions, we also prove Lipschitz continuity and regularity of free interfaces, making use of a parabolic Almgren-type monotonicity formula.

Near BPS skyrmions and restricted harmonic maps

Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map φ:(M,g)→(N,h) between Riemannian manifolds is restricted harmonic if it locally extremizes E2 on its SDiff(M) orbit, where SDiff(M) denotes the group of volume preserving diffeomorphisms of (M,g), and E2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that φ is restricted harmonic if and only if φ∗h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R3→SU(2) and R2→S2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E2 for φ:Rk→N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted F-criticality where F is any functional on maps (M,g)→(N,h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E2 and E4, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R3→S3≡SU(2) is stable restricted F-critical for every such F.

The harmonic map problem with mixed boundary conditions

Given two polygons $S \subset \mathbb {R}^2$ and $\Sigma \subset \mathbb {R}^m$ with the same number of sides, we prove the existence and uniqueness of a smooth harmonic map $u:S \to \mathbb {R}^m$ satisfying the mixed boundary conditions for $S$ and $\Sigma$. This solution is constructed and characterized as a minimizer of the Dirichlet’s energy in the class of maps which satisfy the first mixed boundary condition. Several properties of the solution are established. We also discuss the mixed boundary conditions for harmonic maps defined in smooth domains of the plane.

Some Riemann boundary value problems in Clifford analysis (I)

In this article, we first give the Plemelj formula for functions with parameter by following the classical method. Then by using the higher order Cauchy integral representation formulas, some properties for harmonic functions and bi-harmonic functions are presented, for example, the mean value theorem, the Painlevé theorem, etc. Finally, we consider the Riemann boundary value problems for harmonic functions and bi-harmonic functions in Clifford analysis, the solutions are given in an explicit way.

The harmonic mapping problem and affine capacity

The harmonic mapping problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in the calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents a considerable challenge and leads to several interesting open questions.

Intrinsic Semiharmonic Maps

For maps from a domain Ω⊂ℝm into a Riemannian manifold N, a functional coming from the norm of a fractional Sobolev space has recently been studied by Da Lio and Riviere. An intrinsically defined functional with a similar behavior also exists, and its first variation can be identified with a Dirichlet-to-Neumann map belonging to the harmonic map problem. The critical points have regularity properties analogous to harmonic maps.

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.

On the existence of smooth self‐similar blowup profiles for the wave map equation

AbstractConsider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere).In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self‐similar blowup profile.More generally, we study the relation between the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and the existence of a smooth blowup profile for the (hyperbolic) wave map problem. This has several applications to questions of regularity and uniqueness for the wave map equation. © 2008 Wiley Periodicals, Inc.

Integral-equation method in the mixed oblique derivative problem for harmonic functions outside cuts on the plane

The mixed problem for the Laplace equation outside cuts on the plane is considered. As boundary conditions, the value of the desired function on one side of each of the cuts and the value of its oblique derivative on the other side are prescribed. This problem generalizes the mixed Dirichlet-Neumann problem. By using the potential method, the problem reduces to a uniquely solvable Fredholm integral equation of the second kind.

The harmonic problem in the plane with cuts under boundary conditions of the third kind

The Dirichlet and Neumann problems for harmonic functions in the plane with cuts of any shape were studied in [1] and [2], respectively; the Dirichlet problem was studied by the method of singular integral equations of the first kind, while the Neumann problem was studied by the method of hypersingular integral equations of the first kind. The problem with skew derivatives, whose special case is the Neumann problem, was studied for harmonic functions in the plane with cuts of any shape in [3, 4], where it was reduced to a uniquely solvable Fredholm integral equation of the second kind. Another generalization of the Neumann problem was considered in [5]. The mixed problem for harmonic functions with the Dirichlet condition on a part of the cuts and the condition with a skew derivative, which generalizes the Neumann condition, on the remaining cuts was studied in [6] by using a reduction to a uniquely solvable Fredholm integral-algebraic equation of the second kind in a Banach space. A special case of this problem [6] is the Dirichlet problem in the plane with cuts of any shape. The Dirichlet and Neumann problems in the plane with cuts for the Helmholtz equation were studied in [7, 8], where they were reduced to uniquely solvable Fredholm equations of the second kind. In this paper, we study the boundary value problem for harmonic functions in the plane with cuts under third-kind boundary conditions on both sides of the cuts. We prove the existence and uniqueness of a solution to this problem. We also obtain an integral representation for the solution in the form of potentials. The density in the potentials are determined from a uniquely solvable system of integral equations. The singularities of the gradient of the solution at the endpoints of the cuts are also studied.