# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Polar growth property for harmonic Maass forms

The definition of a harmonic Maass form consists of three properties; (1) that it is modular, (2) that it is harmonic, and (3) that it has at most polar growth at the cusps (ordered in accordance with ...
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### Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
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### Subharmonic function in unbounded regions

The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$: $$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\langle \alpha_i, \alpha_j \rangle$ ...
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### 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 $f_{\epsilon}$ ...
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### 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 ...
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### 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 ...
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 ...