Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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19 views

Difference between a Fourier series and a Harmonic series.

I have read that both these terms are used to mean a superposition of sinusoidal waves or functions with frequencies which are integral multiples of the frequency of the lowest frequency term (also ...
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23 views

Proof of the mean value property

Let $U\subset\mathbb{R}^n$ be an open and connected set and $u\in C^2(U)$ satisfying $-\Delta u=0, x\in U$. Moreover, let $x\in U$ and $r>0$ such that the open ball $B(x,r)\subset U$. Let $h_x(v):=...
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2answers
36 views

Is there any way to calculate harmonic or geometric mean having probability density function?

I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like $K$, I have ...
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16 views

Behavior of Poisson Integral Formula for Half Space

I am trying to solve the following: If $g$ is bounded and $g(x)=|x|$ for $x\in \mathbb{R}^{n-1}$ such that $|x|\leq 1$, then $Du$ is unbounded in a neighboorhood of zero. Remembering that: the ...
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1answer
28 views

Proof for Mean Value Property using a specific limit

I am trying to prove the following: Suppose $u \in C^2(\Omega)$. For some $x \in \Omega$ we have that \begin{align} \Delta u(x) = \lim_{r \to 0} \frac{2n}{r^2} \left[ \frac{1}{w_n} \int_{\...
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1answer
32 views

About polynomial harmonic functions

I am really struggling to solve the following, I don't even know how to start. I would appreciate if anyone could give me some help. Let $m$ be a positive integer and $u : \mathbb{R}^{n} \...
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2answers
58 views

Are harmonic functions always real analytic?

Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb{D}^n$. Let $f$ be a harmonic function w.r.t $g$. Is it true that $f$ must be real-analytic? I think that this ...
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29 views

Laplace problem for a circle with different type of boundary conditions

Let us consider the Laplace problem $\Delta u=0$ with the unitary circle as domain. The boundary conditions are: \begin{cases} u=1 & \quad \text{if } \rho=1 \text{ and } \theta\in[\pi/...
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1answer
33 views

Finding A Function For The Harmonic series [duplicate]

Is it possible to formulate a function that can generate the next number in the harmonic series, for instance: When $$ y = 4,$$ $$x=1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$ Thanks for your ...
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61 views

Summing The Following: $\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$

I can't seem to get this to sum, I'd be very thankful if someone could help me out. $$\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$ NOTE It would be ...
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1answer
156 views

How to compute $\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$

How to compute $$\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$$ where $\operatorname{Li}_2(x)=\sum_{n=1}^\infty \frac{x^n}{n^2}$ is the dilogarithm function. ...
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47 views

How to show a harmonic function is a polynomial?

Can anyone give me a hint to show the following? Let $m$ be a positive integer and $u : \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a harmonic function. If $u(x) = O(\left|x \right|)^m$ when $\left|x \...
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1answer
47 views

Show that $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in the open disc with center 0 and radius 1

Show that the function $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in $K(0,1)$: the open disc with center 0 and radius 1, and the determine the conjugate harmonic functions to $u$. I've been given the ...
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9 views

Local barrier can be extended to a barrier

If $\Omega, \Omega_1$ are bounded, open, connected subsets of $\mathbb{R}^n$ with $\Omega_1\subset \subset \Omega$ (alternatively, $\overline{\Omega_1}\subset \Omega$), $\zeta\in \partial \Omega_1$ ...
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1answer
22 views

$\Delta u = 1$ and $u$ convex $\implies$ $u$ quadratic polynomial

Let $ u \colon \mathbb{R}^{n} \to \mathbb{R} \in C^{4}(\mathbb{R}^{n}) $ be a convex function such that $\Delta u = 1$. Show that $u$ is a quadratic polynomial. What I have done: Every harmonic ...
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6 views

Is a solution of an elliptic PDE constant if all of its derivatives vanish at a point? (in analogy to unique continuation principle)

Consider a one-dimensional elliptic second order differential equation on an interval $[a, b]$. I am specifically interested in Sturm-Liouville problems where the principal symbol is the Laplacian and ...
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1answer
62 views

Laplace equation on the upper half plane with boundary conditions along x=0

Consider the 2d Laplace equation $$\Delta u=0$$ on the upper half plane $$\{(x,y)| y \geq 0\}$$ I know the problem is well posed if you specify Dirichlet boundary conditions at $y=0$, and a suitable ...
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79 views

Counterexample to uniform convergence of Laplace series (expansion in spherical harmonics)

Expansion of a real function on 2-sphere in spherical harmonics, so-called Laplace series, converges uniformly for continuously differentiable functions (see e.g. https://projecteuclid.org/euclid.bbms/...
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Harmonic in punctured ball and bounded implies harmonic in ball.

If $u$ is harmonic and bounded in $B_1(0)\setminus\{0\}$, then can we say that $u$ is harmonic in $B_1(0)$? I believe the answer is yes and I think the way to show it is by the Mean Value Property... ...
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1answer
45 views

Continuous in punctured ball, bounded implies can be continuously extended?

If $n\geq 3$ and $f: B_1(0)\setminus\{0\}:\to \mathbb{R}$ is continuous and bounded, then can $f$ be defined at $x=0$ so that $f$ is continuous on $B_1(0)$? I believe this should be true but I am not ...
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23 views

Harmonic circle-valued maps

Let $M$ be a closed Riemannian manifold. A circle-valued function $u : M \to \mathbb{S}^1$ is harmonic if the associated one form $h_u = u^*(d \theta)$ is harmonic in the Hodge sense: $dh_u = 0$ and $...
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38 views

If $f$ is a holomorphic function then $|f|$ is strictly sub-harmonic.

I got this problem in an exam and it looks so simple. Of course the cauchy’s formula for a holomorphic function $f$ allows one to say that $|f|$ is sub harmonic. But I am not able to see why it should ...
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1answer
91 views

Is the Dirichlet boundary condition continuous?

First, let's fix some notation: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Given a real-valued function $f \in C^{\infty}(\mathbb D^n)$, we denote by $\omega_f$ ...
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16 views

Subharmonic functions inequality

The problem statement Let $u\left( {x,y} \right),v\left( {x,y} \right)$ the solutions of the following problems: $$\left\{ \begin{array}{l} \Delta u = {e^{{x^2} + {y^2}}}{\rm{ \quad\quad ...
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2answers
56 views

If $u$ is harmonic then for $r_1 < r_2$ $ \frac{1}{\text{area}(S_{r1})} \int_{S_{r1}} u^2 dS \le \frac{1}{\text{area}(S_{r2})} \int_{S_{r2}} u^2 dS$

Let $u:\mathbb{R}^3 \to \mathbb{R}$ a harmionic function, and let $S_r = \{x^2+y^2+z^2 = r^2 \}$ I want to show that if $0 < r_1 <r_2$, then: $$ \frac{1}{\mathrm{area}(S_{r_1})} \int_{S_{r_1}} ...
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52 views

show that $f(r) = \frac{1}{\mathrm{area}(S_r)} \int_{S_r} u^2 dS$ is increasing, when $u$ is harmonic

Heading ##Let $u: \mathbb{R}^3 \to \mathbb{R}$ be a harmonic function. For every $r>0$, I denote: $$ f(r) = \frac{1}{\mathrm{area}(S_r)} \int_{S_r} u^2 dS $$ when $S_r = \{(x,y,z)\mid x^2+y^2+z^2 =...
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61 views

Let $u \in C^3(\mathbb{R}^n)$ and harmonic such that $u(x) = o(|x|)$ when $|x| \rightarrow \infty$. Show that $\nabla u(0) = 0$ and u is constant

Let $u \in C^3(\mathbb{R}^n)$ and harmonic such that $u(x) = o(|x|)$ when $|x| \rightarrow \infty$. 1. Show that $\nabla u(0) = 0$ 2. Show that u is a constant function. To show that $\nabla u(0) = 0$...
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1answer
35 views

Harmonic(?) sine and its abbreviation

(I'm not sure about the adjective “harmonic” — maybe it was another one.) Some years ago I read about harmonic sine function (or sine harmonic one) — probably in some physics textbook — where it was ...
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25 views

Why does harmonic imply weakly harmonic?

I am using the following definitions. Let $\Omega$ be a bounded domain in $\mathbb R^d$. A function $v\in L^2(\Omega)$ is weakly differentiable in the $x^i$-direction if there exists a ...
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1answer
39 views

Confusing sign error in proof of Jensen's formula

I'm trying to prove Jensen's formula relating the number of zeros of an entire function to its logarithmic averages on circles. A step in the proof is to show that $$\frac{1}{2\pi} \int_0^{2\pi} \log \...
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1answer
16 views

The imaginary part $~v~$ is equal to $~\frac{1}{2} \log(x+y)~$ . Verify whether it is harmonic or not?

The imaginary part $~v~$ is equal to $~\frac{1}{2} \log(x+y)~$ . Verify whether it is harmonic or not? Relayed to complex functions ie harmonic functions
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Laplace Equation on Rectangle with Inhomogeneous Neumann BCs and PBCs

I will preface this question by referring to a similar question here which did not give a full solution. I found this question on the January 2017 Applied Mathematics Qualifying Exam from the ...
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12 views

Equivalent definition of subharmonicity on a Riemann surface

Let $X$ be a Riemann surface. This is the definition of subharmonicity I have known (and have been trying to work with) for a while. Here is my definition of subharmonic: Let $A$ denote the set of ...
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1answer
34 views

Heuristic argument to prove integrable harmonic function is null

Referring to this question on harmonic function in which is proved the sequent Let $u(x)$ a harmonic function in $\mathbb{R}^n$ such as: \begin{equation} \int_{\mathbb{R}^n}|u(x)|dx =K< \...
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Bounded harmonic function on slit upper half-plane

Let $$\Omega = \{z \in \mathbb{C} : \textrm{Im}(z) > 0\} \setminus \{iy : y \geq 1\}.$$ I need to find a bounded harmonic function $u \colon \Omega \to \mathbb{R}$ such that for each $x \in \mathbb{...
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36 views

If a function $f$ is sufficiently close to a strictly subharmonic function $g$, then $f$ is also strictly subharmonic?

In my setting I'm working on an open subset $Y$ of a Riemann surface, but I don't think that matters too much. I have a smooth function $f: Y \to \mathbb{R}$ that is strictly subharmonic. If another ...
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1answer
46 views

Identity principle for harmonic functions

The question: Show that if $u$ is real and harmonic on a connected open set $U \subseteq \mathbb{C}$, and $ u = 0$ on some small disc $D(P,r) \subseteq U$, then $u = 0$ on all of U. Now here is a ...
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30 views

Holomorphic function $f = u + iv$ on closed punctured unit disk such that $f$ has a non-removable singularity at $0$ and $u$ is harmonic at $0$.

Does such a function $f$ exist? I think the answer is no. My attempt: Take a harmonic conjugate of $u$, say $v'$ so that $g = u + iv'$ is holomorphic on the closed unit disk (not sure what this ...
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1answer
59 views

Book recommendation: compact Riemann surfaces

I am currently reading Jost' Compact Riemann Surface. Can someone recommend some books on the same topic (more specifically, chap. 2-4 of Jost's book), but with more detailed explanations and proofs?...
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Some basic questions about harmoic mapping between two domains

Given a convex $\hat{\Omega}\subset\mathbb{R}^3$, a simply connected domain $\Omega\subset\mathbb{R}^3$, and a boundary correspondence from $\partial\hat{\Omega}$ to $\partial\Omega$, does there ...
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78 views

Existence of Inverse Fourier Transform for a function

For the function (all variables real, and $m>0$) $f(x) = \frac{m}{\sqrt{\lambda }} \tanh \left(\frac{m \sqrt{x^2}}{\sqrt{2}}\right)$, the Fourier Transform is given by (result from Mathematica, ...
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1answer
109 views

Green's identity and gradient estimate

After the proof of the Green's identity in the book "Han Q., Lin F. - Elliptic partial differential equations - AMS (1997)", they state at page 9: We may employ the local version of the Green's ...
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Question about harmonic functions and complex numbers

The question: Give two different harmonic functions on $\mathbb{C}$ that vanish on the entire real axis. Now I don't understand what they mean by vanish. Does that mean $Re(u) = 0$ The simplest ...
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40 views

Find $P(x^{2}-y^{2})$ in terms of $x^{2}-y^{2}$ [closed]

Let $f(z)=P(x^{2}-y^{2})+iQ(x,y)$ be a holomorphic function where P and Q are of class $C^{2}$ I tried to use the fact that $P$ is harmonic , but couldn't continue
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36 views

The topology of the level set of real valued harmonic function through the critical point

Let $u=\Re(z^n)$. Now I want to see the set $$ A=\{z:u(z)=u(0)=0\}=\{z=re^{i\theta}:r^n\cos(n\theta)=0\}. $$ Now I don't know how to prove the following statement (highlighted). Basically why does the ...
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28 views

How do I show that the map is orientation preserving?

Let $f:\mathbb{S}^1\to \mathbb{S}^1$ be an orientation preserving homeomorphism, i.e., the lift $\tilde{f}:\mathbb{R}\to \mathbb{R}$ is monotonic increasing. Now I define $F:\overline{\mathbb{D}}\to \...
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49 views

Holomorphic function on unit disk satisfies $|f(z)| \leq |f(0)| + 2M/(1-|z|)$

I'm having difficulty in solving this question. It comes from a Complex Analysis book, the chapter of Harmonic Functions and Schwarz's Problem on the unit disk Question: Given $f$ a non-constant ...
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1answer
18 views

Proof that the harmonic function defined by the Poisson integral is continuous on the boundary of a ball.

I am reading the text Elliptic Partial Differential Equations of Second Order by D.Gilbard and N.Trudinger, I am struggling with a particular proof they have given. (p.21) In the text they are ...
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36 views

Question about harmonic functions and gradients

Question: Let $(v_1, v_2)$ be a pair of harmonic functions on a disc $U \subseteq \mathbb{C}$. Suppose that $\frac{\partial v_1 }{\partial y} = \frac{\partial v_2}{\partial x}$ and $\frac{\partial ...
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34 views

Laplacian with 1 non-homogeneous Dirichlet, 1 homogeneous Neumann and rest all homogeneous Dirichlet

I need to solve the three-dimensional Laplacian $$\nabla^2T(x,y,z)=0 \tag {A}$$ on a cuboidal domain $x\in[0,a], y\in [0,b], z\in [0,c]$. (A) is subjected to the following boundary conditions $$T(...