Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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2
votes
1answer
94 views

Positive subharmonic functions with constant integral blowing up at boundary

Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying $\Delta f_n\ge 0$ (subharmonic) $f_n\ge 0$ $\int_\Omega f_n=I>0$ ...
7
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1answer
269 views

Harmonic functions on knot complements

In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} =...
4
votes
1answer
142 views

Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
7
votes
0answers
177 views

“Universal” polynomial of bounded norm on the sphere

Consider the space $V_{d,n}=\mathbb{R}[x_1,\ldots,x_n]_d$ of homogeneous polynomials of degree $d$ in $n$ variables. I am interested in the set of bounded polynomials on the sphere $$B_{d,n}=\{f\in V_{...
1
vote
1answer
82 views

Subharmonic function in unbounded regions

The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$: $$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
4
votes
1answer
84 views

Periods of the harmonic conjugate and a Dirichlet problem on a multiply connected domain

Any harmonic function $u$ on a simply connected domain in $\mathbb{R}^2$ is the real part of a holomorphic function. If the domain is multiply connected, then this is no longer true: the harmonic ...
1
vote
1answer
147 views

A question of uniqueness

Let $u$ an harmonique function on $\Omega=(a,b)\times (0,+\infty)$ and boundary conditions : $\displaystyle u(a,y)=u(b,y)=0,\quad\forall y\geq 0$ $\displaystyle u(x,0)=0,\,\lim_{y\to +\infty} u(x,y)=...
5
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1answer
275 views

A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$. What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
1
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0answers
63 views

how to construct a finite energy map

In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...
0
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0answers
23 views

Reference on area-constrained harmonic maps?

I would appreciate a reference that discusses (at least some of) what is known regarding the existence of constrained harmonic maps from one space into another. More specifically, I am interested in ...
1
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1answer
81 views

Approximation of smooth function by harmonic function

Let $u(x,y)$ be a smooth function on the square $S:=[0, 1] \times [0, 1]$ (see, for example, Wiki for the definitions) and $\varepsilon > 0$. Is it possible to approximate $u(x,y)$ by a function $...
5
votes
1answer
202 views

Every homotopy class contains at least a harmonic representative

Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...
2
votes
1answer
120 views

When is this differential form harmonic?

Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions ...
4
votes
1answer
80 views

Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives

Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary). Does there exist a sequence of ...
6
votes
2answers
594 views

A harmonic function

Consider the vertical strip of angle $\alpha=\frac{\pi}{2}$ In this case, the harmonic function which is $0$ on the left line and $1$ on the right line is given by $$f(a+ib)=\frac{a}{T}.$$ Now, when ...
1
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0answers
49 views

Local properties of harmonic forms on riemannian manifolds

Consider a riemannian manifold together with a orthogonal basis $\{\alpha_1,\dots,\alpha_n\}$ of the space of harmonic $k$-form. I suspect that the inner product $\langle \alpha_i, \alpha_j \rangle$ ...
1
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1answer
78 views

Smooth approximation of a subharmonic function in the barrier sense

Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f_{\epsilon}$ ...
4
votes
1answer
120 views

an Integral Inequality about harmonic function

Let $u$ be a real-valued harmonic function on $\mathbb{D}$, which extends continuously to the boundary. I wonder how to prove the inequality $$\int_\mathbb{D} e^{2u} dxdy\leq\dfrac{1}{4\pi}\left(\int_{...
5
votes
1answer
168 views

Can harmonic maps with immersive boundary conditions have singular points?

Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
2
votes
1answer
100 views

regularity of p-harmonic functions

We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$. Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
0
votes
3answers
102 views

Intersection of Subharmonic and Harmonic Functions (also: Extension of harmonic functions)

Let $\Omega \subset \mathbb{R}^n$ be open, convex and bounded with smooth boundary. Define $$\mathcal{A}(\Omega) = \left\{ C \subset \Omega \ \big\vert \begin{array}{l} \text{for any subharmonic ...
2
votes
1answer
148 views

The positive solutions of the weighted Laplacian equation

Let $u$ be a positive function on $\mathbb R^n$ such that $$ \Delta u-\partial_{x_1}u=0, $$ where $\Delta$ is the Laplacian operator $\partial_{x_1}^2+\partial_{x_2}^2+\cdots+\partial_{x_n}^2$. Can ...
8
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0answers
106 views

implicit function theorem and harmonic mapping

We are given two Riemannian manifolds $M,N$ of dimension $m$ and $n$ and a function $G \colon M \times N \to \mathbb{R}^n$ which satisfies the assumptions of the implicit function theorem, meaning ...
1
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1answer
215 views

Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...
5
votes
0answers
126 views

Continuity of the Green function with respect to the measure

Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as $$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$ where $\mu^{*n}$ is the $n$th convolution power of $\...
1
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0answers
71 views

Existence of nonparabolic ends

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...
0
votes
1answer
95 views

How to create a function whose harmonic is a sine wave [closed]

How do I solve the following equation for $f(\cdot)$? $f(x)+\frac{1}{n}f(nx)=\sin(x)$ That is, how do I create a function which, when combined with its nth harmonic, will be a sine wave?
5
votes
1answer
169 views

Taylor-like expansion for a holomorphic function in non-simply-connected domain

Suppose $f$ is a holomorphic function in a simply connected open set $U$, and we know it's Taylor expansion at a point $p\in U$. We can then find a holomorphic map $g$ of $U$ to the unit disc which ...
4
votes
0answers
66 views

Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...
2
votes
0answers
119 views

Motivation and examples of parabolic manifolds

Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...
0
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0answers
72 views

On a property of harmonic functions of stochastic processes

I have a question which relates to an argument appearing in this paper 1. Let $D$ be a domain of $\mathbb{R}^d$ and $X=(X_t, P_x)$ be a diffusion process on $D$. Let $h : D \to [0,\infty]$ be a ...
5
votes
1answer
125 views

A question about integration of spherical harmonics on $(S ^ 2, can)$

Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that $$ \int_{\mathbb{...
5
votes
1answer
183 views

harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
4
votes
0answers
156 views

Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
0
votes
0answers
90 views

Hitting probabilities and harmonic functions

I have a question about harmonic functions and hitting probabilities. Let $d \ge 3$ be an integer. Let $D=\mathbb{R}^d \times (-1,1)$ and denote points $z \in D$ by $z=(r,\theta,y),$ where $(r,\...
7
votes
2answers
1k views

Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
5
votes
1answer
134 views

Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell: Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form? Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
1
vote
0answers
92 views

Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{...
2
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0answers
52 views

Does a map which preserve harmonic forms preserve co-closed forms (locally)?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth. Let $1 \le k \le d-1$ be fixed....
2
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1answer
1k views

Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
8
votes
2answers
389 views

Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist a ...
2
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0answers
84 views

On the solution of Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system $$ \begin{...
4
votes
1answer
99 views

Dynamics for approximating harmonic functions on graphs

A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors. Consider the following method for approximating a ...
1
vote
1answer
97 views

Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field

Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
0
votes
1answer
106 views

Harmonicity of the Martin kernels

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum_{n\geq 0}\mu^{*n}(x^{-1}y)$, where $\mu^{*n}$ is the $n$th ...
1
vote
1answer
135 views

Topological similarity of solutions to Dirichlet problem

Let $\varphi_{1},\varphi_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two smooth general position (Morse) functions having the same set of critical points $\left\{ p_{1},...,p_{n}\right\} \subset\...
4
votes
1answer
125 views

A question about harmonic function

Let $u$ be a harmonic function inside the unit ball $B(0, 1)$ in $\mathbb{R}^2$ so that $|u|\leqslant 1$. Does a function u which satisfies $|\nabla u(0)|>1$ exist? If not, please prove that $|\...
1
vote
1answer
285 views

Functions orthogonal to harmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose $\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
1
vote
0answers
54 views

Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?

Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values? If the answer is negative then we conclude ...
12
votes
2answers
1k views

Harmonic function properties on $\mathbb R^3$

Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...