Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
1,699
questions
0
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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^...
0
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0answers
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:
$$\frac{\partial^2}{\partial x^2} f(x,y) +...
0
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0answers
8 views
Relation between Poisson kerner and Green functions in disk
Recently I have found in some lecture notes the following intriguing relation that connects Poisson kerner and Green functions on disk,
$$P(w,\zeta)=\frac{1-|w|^2}{|\zeta-w|^2}=\lim\limits_{z\...
1
vote
0answers
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:...
1
vote
0answers
17 views
$ \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 $u\left( x\right) =0$ ...
1
vote
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 ...
1
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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
$$\Psi(\mathbf{s}) - \frac{m}{2s^2} \frac{k}{2 \pi^2 \hbar} \int_{s S^3} \frac{\Psi(\...
0
votes
1answer
27 views
Can the harmonic function $u$ in $U$ be extended continuously to $\overline{U}$?
Suppose that
\begin{equation}
u(z)=\arg \frac{1+z}{1-z}
\end{equation} is harmonic in the unit disk $U$. Can this function be extended to a continuous real function on $\overline{U}$? I guess it can ...
0
votes
0answers
30 views
General form for differentiable function satisfying mean-value property?
Let $f(x,y)$ be a differentiable and integrable function defined in a domain $D\subset\mathbb{R}^2$.
$\forall(x_0,y_0)\in D$, $C$ is a circle in $D$ with center $(x_0,y_0)$ and arbitrary radius $r$. ...
1
vote
1answer
33 views
Integral over fundamental solution in Green's formula
Consider Green's representation formula
\begin{equation}
u(x) = \int_{\Omega} \Gamma(x-y)\left(-\Delta u(y)\right)\mathrm{d}y+\oint_{\partial\Omega}\Big(\Gamma(x-y)\nabla u(y)-u(y)\nabla_y(\Gamma(x-y))...
0
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1answer
23 views
Directed bipartite graphs with weighted edges with uniform input weight
Here is the set-up:
Let $G = (V(G), E(G))$ be a bipartite graph with $ V_1 \sqcup V_2$ denoting the vertices with respect to 2-colouring of the graph. Consider the induced directed graph $G' = (V(G),A(...
0
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0answers
19 views
Harmonic function with Dirichlet and Neumann boundary
Suppose $D^+_1 \subset \mathbb{R}^2$ is the upper half part of the unit disk and $u$ solves
$$\left\{
\begin{aligned}
\Delta u &=0&
\quad &\text{ in $D^+_1$};\\
\partial_2 u&= f(x,u)&...
0
votes
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 ...
4
votes
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 \...
4
votes
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\...
0
votes
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?
0
votes
0answers
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},|...
1
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0answers
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}\...
1
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0answers
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,...
0
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0answers
19 views
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:
$$u_x=v_y \...
1
vote
1answer
46 views
Elliptic regularity estimate for Possion with bounded forcing term
Suppose I have $- \Delta u = f $ weakly on a bounded domain $\Omega$ with Dirichlet boundary conditions. I'm looking for an elliptic regularity result of the form $f \in L^\infty \implies u \in C^{1, \...
0
votes
0answers
38 views
$f:=u_x -iu_y$ holds the Cauchy-Riemann equations
Let $u: U \subset \Bbb C \to \Bbb R$ be harmonic. We define $f:=u_x -iu_y$. I wanna show that $f$ satisfies the Cauchy-Riemann equations but I am not sure.
I should proof for the first equation that $(...
4
votes
1answer
65 views
Reference request on spherical harmonics
I'd like to find a (hopefully modern, mathematician-friendly) reference which proves that homogeneous harmonic polynomials restrict to an orthonormal basis for $L^2$ functions on the sphere $S^n$. ...
1
vote
1answer
28 views
minimal graphs converge to a harmonic function
Suppose $\{u_i\}$ is a sequence of positive $C^2$-functions on the unit ball $B,$ satisfying the minimal surface equation, i.e.,
$${\rm div}\left(\frac{Du_i}{\sqrt{1+|Du_i|^2}}\right)=0$$
for all $i.$ ...
2
votes
1answer
33 views
Show that $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic if $u$ is harmonic
I have a "simple" question but I'm not able to solve it.
Suppose that $u$ is a harmonic function on $\mathbb{R}^3$. Prove that the function $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic on $\...
0
votes
1answer
31 views
mean value theorem of harmonic function
The mean value theorem of harmonic function tells us: If $U \subset \Bbb R^n$ open, $f: U \to \Bbb R$ harmonic then for all $y \in U$ and $r>0$ s.t. $\overline{B}(y,r) \subset U$ the following is ...
1
vote
0answers
41 views
Estimates for partial derivatives of harmonic functions
I'm interested in proving the following:
Let $u$ harmonic in $\Omega $ and let $ B_{\rho} (x) \subset {\Omega} $ an open ball, then
$$
\left |{ u_{x_{i}}(x) }\right | \leq{ \...
1
vote
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, ...
6
votes
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 ...
0
votes
0answers
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
$$ \frac 1 {2 \pi} \int_{-...
1
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0answers
26 views
Integral solution for Laplace equation in multiply connected domain?
Let $ \Omega \subseteq \mathbb{R}^2 $ be a bounded and simply connected domain with its boundary $\partial \Omega \in C^2 $, $f \in C(\partial \Omega) $ be some given function, then for
$$
\Delta u = ...
0
votes
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:
$$ \left( r^2\frac{\partial^2}{\partial r^2} + r\frac{\partial}{\partial r} + \frac{\partial^2}{\...
0
votes
0answers
17 views
elliptic PDE and operators
Studying now PDE, especially elliptic equations and operators, I do not understand the importance in elliptic operators (especially Laplacian) more than other operators, what does elliptic operators ...
7
votes
2answers
116 views
Find all entire functions $f$ such that $|f|$ is harmonic.
I'm trying to find all entire functions $f$ such that $|f|$ is harmonic. My attempt is as follows.
Because $f$ is entire, we may write $f(z) = f(x + iy) = u(x,y) + iv(x,y)$ where $u$ and $v$ have ...
6
votes
0answers
172 views
Solid harmonic addition theorem in higher dimensions?
The solid harmonics are solutions to Laplace's equation in spherical coordinates. The regular and irregular solid harmonics, obtained by rescaling spherical harmonics, are respectively
$$R_l^m(\textbf{...
0
votes
0answers
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 ...
0
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0answers
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 $...
1
vote
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^...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
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0answers
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 ...
0
votes
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
$$
\|Du\| _{L^\infty(\Omega')} \leq \frac{N}{d(\Omega', \partial \...
0
votes
0answers
27 views
Given a function $f(z)$ that is analytic over the whole complex plane and Im($f$) $\leq 0$, show that $f$ is constant
By according to the Maximum Principle for Harmonic Functions, I could say that Im($f$)$= v$ is constant and, so, by using the Cauchy-Riemann equations, I prove that $f$ is constant over the whole ...
0
votes
0answers
39 views
Given a smooth function, does there exist a (semi-)Riemannian metric for which it is harmonic?
To be precise: given $\phi \in C^{\infty}(M)$, does there exist a (semi-)Riemannian metric $g$ such that $\Delta_{g}\phi = 0$?
I avoided fixing a coordinate system on purpose here; I'm not sure how ...
1
vote
0answers
16 views
How to find the complex function corresponding to a potential function?
I have a potential function $F(x,y)$ that satisfies the 2D Laplacian $\nabla^2 F = \frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2} =0$. Given the property of potential functions, a ...
1
vote
1answer
65 views
If $u$ is harmonic in $B(0,1)\subset\mathbb R^2$ with $\vert u\vert\leq1,$ then $\vert\nabla u(0)\vert\leq4/\pi$
Let $u$ be harmonic in $B(0,1)\subset\mathbb R^2$ and $\vert u\vert\leq1,$ then $\vert\nabla u(0,0)\vert\leq4/\pi.$
Since $\vert\nabla u(0,0)\vert=\sqrt{u_{x}(0,0)^2+u_{y}(0,0)^2}$, we need to ...
1
vote
1answer
70 views
Equivalent definitions of subharmonicity (Mean Value Inequality vs Viscosity vs Classic)
Consider the following problem:
Let $\Omega \subset \mathbb R^N$ be a domain and $u \in C(\Omega)$. Show that $u$ is subharmonic in the Mean Value Inequality sense, that is,
$$
u(x) \leq \frac{1}{\...
0
votes
0answers
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 ...
0
votes
0answers
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 $g(\mathbf{x},y) = \frac{y}{(|| \mathbf{x} || ^{2} + y^{2})^{(n+1)/2}}$ ...
