Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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1answer
13 views

Supremum of a subharmonic function

Le 𝐷 be an open ball of $\mathbb{R}^2$, 𝑓:$\mathbb{R}^2→\mathbb{R}$ a function and $\Delta$ the Laplacian operator. $m_1$ is a local maximum of $f$ in $D$. Recall that $\forall h \in \mathbb{R}^2, d^...
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16 views

Solution of Laplace equation in an annulus?

Let the annulus be a 2D region between two concentric circles with inner circle radius $R_1$ and outer circle radius $R_2$. How to solve the Laplace equation: $$\frac{\partial^2}{\partial x^2} f(x,y) +...
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8 views

Relation between Poisson kerner and Green functions in disk

Recently I have found in some lecture notes the following intriguing relation that connects Poisson kerner and Green functions on disk, $$P(w,\zeta)=\frac{1-|w|^2}{|\zeta-w|^2}=\lim\limits_{z\...
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33 views

Holomorphic function with given boundary values

I have a complex valued function on the boundary of a set, and want to know if it has a holomorphic extension to the entire set (holomorphic in interior, continuous up to the boundary). In other words:...
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$ \lim_{x\rightarrow0}\left\vert x\right\vert ^{n-2}u\left( x\right) =0. $ Show that $u\equiv0$ in $B_{1}\left( 0\right) $.

Let $n\geq3$, $B_{1}\left( 0\right) \subset\mathbb{R}^{n}$ and $u\in C\left( \overline{B_{1}\left( 0\right) }\backslash\left\{ 0\right\} \right) $ be harmonic. Suppose $u\left( x\right) =0$ ...
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0answers
26 views

Dirichlet energy given Dirichlet boundary condition

Let $\Omega = B(0,1) \subset \mathbb R^n$. Given $f \in C(\partial \Omega)$ (though I guess what I want to do may make more sense in some other space of functions), we can solve the Dirichlet problem ...
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0answers
26 views

Laplace equation on the 3-sphere

In the article The Ellipse and the Atom, the author mentions, but does not precisely prove, that the equation $$\Psi(\mathbf{s}) - \frac{m}{2s^2} \frac{k}{2 \pi^2 \hbar} \int_{s S^3} \frac{\Psi(\...
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1answer
27 views

Can the harmonic function $u$ in $U$ be extended continuously to $\overline{U}$?

Suppose that \begin{equation} u(z)=\arg \frac{1+z}{1-z} \end{equation} is harmonic in the unit disk $U$. Can this function be extended to a continuous real function on $\overline{U}$? I guess it can ...
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30 views

General form for differentiable function satisfying mean-value property?

Let $f(x,y)$ be a differentiable and integrable function defined in a domain $D\subset\mathbb{R}^2$. $\forall(x_0,y_0)\in D$, $C$ is a circle in $D$ with center $(x_0,y_0)$ and arbitrary radius $r$. ...
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1answer
33 views

Integral over fundamental solution in Green's formula

Consider Green's representation formula \begin{equation} u(x) = \int_{\Omega} \Gamma(x-y)\left(-\Delta u(y)\right)\mathrm{d}y+\oint_{\partial\Omega}\Big(\Gamma(x-y)\nabla u(y)-u(y)\nabla_y(\Gamma(x-y))...
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1answer
23 views

Directed bipartite graphs with weighted edges with uniform input weight

Here is the set-up: Let $G = (V(G), E(G))$ be a bipartite graph with $ V_1 \sqcup V_2$ denoting the vertices with respect to 2-colouring of the graph. Consider the induced directed graph $G' = (V(G),A(...
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19 views

Harmonic function with Dirichlet and Neumann boundary

Suppose $D^+_1 \subset \mathbb{R}^2$ is the upper half part of the unit disk and $u$ solves $$\left\{ \begin{aligned} \Delta u &=0& \quad &\text{ in $D^+_1$};\\ \partial_2 u&= f(x,u)&...
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2answers
23 views

Problem on real-valued harmonic function.

Problem: Let $g$ be harmonic on $U$ where $U \subseteq \mathbb{R}^2$ is open and connected. Show that if $g$ is non-constant on $U$, then $g(U)$ is open in $\mathbb{R}$. I noticed that we could show ...
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2answers
104 views

Harmonic functions $u$ and $v$ such that $\nabla u = a \nabla v$ for some function $a$.

Let $u,v\colon\Omega\to\mathbb R$ be harmonic functions on the open set $\Omega \subset\mathbb R^2$ such that: Exists function $a\colon\Omega\to\mathbb R$ such that $\nabla u \left(x,y\right) = a \...
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1answer
80 views

Unique continuation at the boundary for harmonic functions in the plane

Consider the set $U = (-1,1) \times \{ 0\} \subset \mathbb R^2$ and a continuous function $f : U \rightarrow \mathbb R$. Then for any domain $\Omega \subset \mathbb R^2$ such that $U \subset \partial\...
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1answer
19 views

Without calculating the partial derivatives, explain why sin x cosh y and cos x sinh y are harmonic functions in C.

It could be simple if I was allowed to use partial Derivatives. Without calculating partial derivative I don't know how to prove that it's harmonic. Can someone please amswer this question?
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21 views

Finding bounded harmonic function on upper half-plane with Dirichlet boundary conditions.

I have been asked to find a bounded function $u(z)$ that is harmonic on $\Omega= \{z\in\mathbb{C}|\,|z|>1\,,\Im(z)>0\}$ that satisfies \begin{equation} u(z)=\begin{cases} 1\quad z\in\mathbb{R},|...
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12 views

BMO/BMOA spaces

I am working with $BMO$ and $BMOA$ spaces (Let us assume in the disk). I am asking myself if it is true that $BMO=BMOA + \bar{BMOA}$, where with $\bar{BMOA}$ , I mean functions in $BMOA(\mathbb{C}\...
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25 views

If for an harmonic function $u$ in $\mathbb R^n \times \mathbb R_+$, $u(\mathbf{0},y) = 0$ then $u \equiv 0$?

If $u$ is a harmonic function on $\mathbb R^n \times \mathbb R_+$ and $$u(\mathbf{0},y) = 0$$ (here $\mathbf{0} \in \mathbb R^n$ and $y \in \mathbb R_+$) and $u$ depends radially on the first variable,...
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Harmonic function is the real part of a holomorphic function [duplicate]

Let $u: \Bbb R^2 \to \Bbb R$ be an harmonic function, then there exists an holomorphic function $f: \Bbb C \to \Bbb C$ s.t. $Re(f)=u$. The only hint I have are the Cauchy–Riemann equations: $$u_x=v_y \...
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1answer
46 views

Elliptic regularity estimate for Possion with bounded forcing term

Suppose I have $- \Delta u = f $ weakly on a bounded domain $\Omega$ with Dirichlet boundary conditions. I'm looking for an elliptic regularity result of the form $f \in L^\infty \implies u \in C^{1, \...
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38 views

$f:=u_x -iu_y$ holds the Cauchy-Riemann equations

Let $u: U \subset \Bbb C \to \Bbb R$ be harmonic. We define $f:=u_x -iu_y$. I wanna show that $f$ satisfies the Cauchy-Riemann equations but I am not sure. I should proof for the first equation that $(...
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1answer
65 views

Reference request on spherical harmonics

I'd like to find a (hopefully modern, mathematician-friendly) reference which proves that homogeneous harmonic polynomials restrict to an orthonormal basis for $L^2$ functions on the sphere $S^n$. ...
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1answer
28 views

minimal graphs converge to a harmonic function

Suppose $\{u_i\}$ is a sequence of positive $C^2$-functions on the unit ball $B,$ satisfying the minimal surface equation, i.e., $${\rm div}\left(\frac{Du_i}{\sqrt{1+|Du_i|^2}}\right)=0$$ for all $i.$ ...
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1answer
33 views

Show that $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic if $u$ is harmonic

I have a "simple" question but I'm not able to solve it. Suppose that $u$ is a harmonic function on $\mathbb{R}^3$. Prove that the function $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic on $\...
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1answer
31 views

mean value theorem of harmonic function

The mean value theorem of harmonic function tells us: If $U \subset \Bbb R^n$ open, $f: U \to \Bbb R$ harmonic then for all $y \in U$ and $r>0$ s.t. $\overline{B}(y,r) \subset U$ the following is ...
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41 views

Estimates for partial derivatives of harmonic functions

I'm interested in proving the following: Let $u$ harmonic in $\Omega $ and let $ B_{\rho} (x) \subset {\Omega} $ an open ball, then $$ \left |{ u_{x_{i}}(x) }\right | \leq{ \...
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0answers
26 views

Beltrami equation with harmonic coefficient.

I need to find solutions to the Beltrami equation $$ \frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z} $$ for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
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2answers
130 views

Is every harmonic polynomial a linear combination of these?

In $N$-dimensional space, we can show by direct calculation that the polynomial $$ r^{2K+N-2}\nabla_a\nabla_b\nabla_c\cdots \frac{1}{r^{N-2}} \hspace{1cm} \text{(with $K$ derivatives)} $$ is ...
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37 views

Find an integration kernel related to the Poisson kernel

$\DeclareMathOperator\Log{Log}$ $\DeclareMathOperator\Arg{Arg}$ Problem statement I'm trying to find an integration kernel $K_r(t)$ (ie: doesn't depend on $\alpha$) such that $$ \frac 1 {2 \pi} \int_{-...
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0answers
26 views

Integral solution for Laplace equation in multiply connected domain?

Let $ \Omega \subseteq \mathbb{R}^2 $ be a bounded and simply connected domain with its boundary $\partial \Omega \in C^2 $, $f \in C(\partial \Omega) $ be some given function, then for $$ \Delta u = ...
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2answers
28 views

change of variable to get a quasi-cartesian laplace equation?

when writing the (vector) laplace equation in cylindrical coordinates in a $(r,\theta)$ plane, we get: $$ \left( r^2\frac{\partial^2}{\partial r^2} + r\frac{\partial}{\partial r} + \frac{\partial^2}{\...
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17 views

elliptic PDE and operators

Studying now PDE, especially elliptic equations and operators, I do not understand the importance in elliptic operators (especially Laplacian) more than other operators, what does elliptic operators ...
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2answers
116 views

Find all entire functions $f$ such that $|f|$ is harmonic.

I'm trying to find all entire functions $f$ such that $|f|$ is harmonic. My attempt is as follows. Because $f$ is entire, we may write $f(z) = f(x + iy) = u(x,y) + iv(x,y)$ where $u$ and $v$ have ...
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172 views

Solid harmonic addition theorem in higher dimensions?

The solid harmonics are solutions to Laplace's equation in spherical coordinates. The regular and irregular solid harmonics, obtained by rescaling spherical harmonics, are respectively $$R_l^m(\textbf{...
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20 views

How to construct such a harmonic function

Construct a harmonic function $u(z)$ in the disc $|z|<R$ for which $\lim_{r\to R} u( r e^{i\theta} )= 0$ if $ 0< \theta <\pi $ and $1$ if $\pi < \theta < 2 \pi$. This question is ...
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10 views

Proving this result for harmonic function for |z|>R which is similar to Integral formula for harmonic functions in the disc |z|<R

This question is from Ponnusamy and silvermann pg 369 and section harmonic functions and I don't know how to prove it. If u(z) is harmonic for |z|>R and continuous for $|z|\geq R$ , show that for $...
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3answers
79 views

How to prove $\int_{0}^{2\pi } \frac{ \sin(\theta - \phi) } { R^2 - 2rR \cos(\theta -\phi) +r^2 } d{\phi} =0$

This question was part of mock test of masters exam for which I am preparing and I am unable to solve it. Show that $\int_{0}^{2\pi } \frac{ \sin(\theta - \phi) } { R^2 - 2rR \cos(\theta -\phi) +r^...
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1answer
43 views

How to use mean value theorem For harmonic functions to prove this

The following question was asked in my complex analysis assignment and I am confused about how this should be done. Show that $\int_{0}^{\pi} \ln( \sin {\theta}) d{\theta} = -\pi \ln 2$ by applying ...
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1answer
21 views

If $u(z)$ is non-constant and harmomic in the plane, then show that $u(z)$ comes arbitrarily close to every real value

This question was asked in my complex analysis assignment and I am not able to solve it. If $u(z)$ is non constant and harmonic in the plane, then show that $u(z)$ comes arbitrarily close to every ...
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1answer
21 views

Growth Rate of Integral of Harmonic functions

This problem comes from a statement in the book $\textit{Elliptic Partial Differential Equations: Second Edition}$ written by Han and Lin. In its chapter 1, there's Lemma 1.41 (in the page 22) as ...
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45 views

Volterra series analysis for nonlinear circuits

I am trying to analyze a nonlinear circuit shown below. Where $R_1$ is a linear component, $1 \Omega $ resistor. And $R_2$ is a non-linear component, a non-linear resistor. Its current changes ...
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1answer
34 views

Gradient estimate for harmonic functions - Isn't a power missing on $N$?

In class we have seen that if $u$ is harmonic, $\Omega \subset \mathbb R^N$ and $\Omega' \subset \subset \Omega$ is a subdomain, then $$ \|Du\| _{L^\infty(\Omega')} \leq \frac{N}{d(\Omega', \partial \...
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0answers
27 views

Given a function $f(z)$ that is analytic over the whole complex plane and Im($f$) $\leq 0$, show that $f$ is constant

By according to the Maximum Principle for Harmonic Functions, I could say that Im($f$)$= v$ is constant and, so, by using the Cauchy-Riemann equations, I prove that $f$ is constant over the whole ...
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39 views

Given a smooth function, does there exist a (semi-)Riemannian metric for which it is harmonic?

To be precise: given $\phi \in C^{\infty}(M)$, does there exist a (semi-)Riemannian metric $g$ such that $\Delta_{g}\phi = 0$? I avoided fixing a coordinate system on purpose here; I'm not sure how ...
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0answers
16 views

How to find the complex function corresponding to a potential function?

I have a potential function $F(x,y)$ that satisfies the 2D Laplacian $\nabla^2 F = \frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2} =0$. Given the property of potential functions, a ...
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1answer
65 views

If $u$ is harmonic in $B(0,1)\subset\mathbb R^2$ with $\vert u\vert\leq1,$ then $\vert\nabla u(0)\vert\leq4/\pi$

Let $u$ be harmonic in $B(0,1)\subset\mathbb R^2$ and $\vert u\vert\leq1,$ then $\vert\nabla u(0,0)\vert\leq4/\pi.$ Since $\vert\nabla u(0,0)\vert=\sqrt{u_{x}(0,0)^2+u_{y}(0,0)^2}$, we need to ...
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1answer
70 views

Equivalent definitions of subharmonicity (Mean Value Inequality vs Viscosity vs Classic)

Consider the following problem: Let $\Omega \subset \mathbb R^N$ be a domain and $u \in C(\Omega)$. Show that $u$ is subharmonic in the Mean Value Inequality sense, that is, $$ u(x) \leq \frac{1}{\...
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15 views

Statistical methods in order to check replicability of datasets

I am facing the following difficulty: I collected several data about stock markets in order to create portfolios for a specific number countries. Now, let's take USA as an example: Let's suppose, I ...
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12 views

Showing function is harmonic

I'm now learning basic real analysis and in some troubles. Let $g : \mathbb{R}^{n} \times (0,\infty) \rightarrow \mathbb{R}$ by $g(\mathbf{x},y) = \frac{y}{(|| \mathbf{x} || ^{2} + y^{2})^{(n+1)/2}}$ ...

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