Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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16 views

Analytic and harmonic functions in the unit disc (Question 4.1.1 of “Complex Polynomials” by Sheil-Small)

The question (not homework) is Let $f\in\mathcal{H}$ and suppose that $f(0) = 0$ and $\left|f(z)\right|\leq 1$ for $z\in\mathbb{U}$. Show that $$\left|f(z)\right|\leq \frac{2}{\pi} arg\left(\frac{1+...
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Deriving (using Fourier transform) the Poisson kernel for solving the Dirichlet problem on unit balls

Let's first consider the following Dirichlet problem on the upper half-space $\mathbb H^n=\{(x_1,\ldots, x_n)\in \mathbb R^n:x_n>0\}$. $$ \Delta u =0, u|_{x_n=0}=g(x). $$ Performing Fourier ...
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Where to find reference for the energy method $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm}$

Why do I need to multiply by the function w in the energy method to guaranty at most one solution? This is the example $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm} u=0 \hspace{0....
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Help with this exercise related to Poisson Kernel and Poisson Integral [closed]

Let $Q_n(t)$ a sequence of even functions, 2 $\pi$ periodic and piecewise on $(-\pi,\pi)$ such that: $Q_n(t) \geq 0$ and $\int_{-\pi}^{\pi} Q_n(t)dt = 1$ For every $\delta > 0, Q_n(t)$ converges ...
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Is it possible to find a generalized mean value equality for harmonic function on an arbitrary domain $\Omega$?

Mean value theorem of harmonic function states that for $u\in C(\overline \Omega)\cap C^2(\Omega)$, and $B(x_0,r)\subseteq \Omega$, where $\Omega \subseteq \mathbb R^n$ is connected and open, we have ...
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1answer
35 views

Is it true that $u$ is harmonic if and only if $u$ satisfies the mean value property?

I know that $u$ harmonic $\implies u$ satisfies the mean value property, but does this work the other way around? Also, if we have Laplace's equation defined inside a disc of radius $1$ with boundary ...
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55 views

Prove that $u(x,y) = \frac{x}{x^2+y^2}$ is harmonic in $\mathbb{R}^2\setminus\{(0,0)\}$

I want to check if I did this right. I reached the conclusion that $u$ is not harmonic. We know that a function is harmonic if $$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\...
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Intuitive explanation of solution to Laplace's equation on unit disc with boundary conditions (i) $u(1,\theta)=c$, & (ii) $u(1,\theta)=\sin(\theta)$?

I know it can be explicitly shown that for (i) $u(r,\theta)=c$, and for (ii) $u(r,\theta)=r\sin(\theta)$ by separating variables and finding the Fourier coefficients, but is there any mathematical ...
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Poisson kernel equivalence

I'm currently reading "Real and Complex Analysis" by Rudin. It was shown that $$ \sum_{n = -\infty}^{\infty} r^{|n|} e^{in \theta} = \frac{1 - r^2}{1 - 2r \cos{\theta} + r^2 } \quad 0 \leq r < ...
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Show function $u \gt 0$ [closed]

$\Omega$ is a bounded area in $R^3$ with $\Gamma$ as bound, function $u$ is the solution of \begin{equation} \left\{ \begin{array}{lr} -\Delta u + cu=f, & c\gt 0, f \gt ...
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70 views

Show u(r, θ) is a solution to the Dirichlet Problem for the unit disk

Show that $u(r,\theta) = \frac{1}{\pi}\arctan\left(\frac{1-x^2-y^2}{(x-1)^2+(y-1)^2-1}\right)\\$ where $\arctan(t) \in [0,\pi]$ is the solution to Dirichlet's problem for a unit disk for the piecewise ...
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1answer
21 views

Harmonic functions theorem

I have to prove this theorem but monstrous computation appear when i try to compute the laplacian of $\hat{u}$. Does anyone know an easier way for the proof? Let be $\Omega$ an open subset in $\...
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Solution using separation of variables of laplace equation

Analytical solution of Laplace equation $$u_{xx} + u_{yy} = 0$$ with boundary condition $$u(x, 0) = 0$$ $$u(x, 1) = 10$$ $$u(0, y) = 0$$ $$u(1, y) = 10$$ Using separation of variable, Suppose $$u(x,y)...
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$2\log |f(z)|-2\log |z|$ is harmonic function where $f$ is holomorphic in an annulus

Let $U$ denote the open annulus $\{z\in \Bbb C:1<|z|<2\}$ and suppose $f:U\to U$ is a holomorphic bijection. I want to show that $u(z)=2\log |f(z)|-2\log |z|$ is a harmonic function on $U$. ...
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Critical points of harmonic functions

Let $(M^3,g)$ be a compact Riemannian manifold with boundary $\partial M \neq \emptyset$, and consider a non constant harmonic function $f : M \to \mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$ satisfying ...
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If $\lim_{r \to 1} \frac{1}{2\pi}\int_0^{2\pi} \left| \log \left| f(re^{it}) \right| \right| dt = 0$ then $\left| f(z) \right| \leq 1$

My aim is to prove that Blaschke products are the only holomorphic funcions that verify the property $$ \lim_{r \to 1} \frac{1}{2\pi} \int_0^{2\pi} \left| \log \left| f(re^{it}) \right| \right| dt = ...
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Green's Function for Dirichlet problems

I have been studying Green's functions for Laplace/Poisson's equation and have been having some trouble on a few things. In Strauss's book he claims the solution to the Dirichlet problem is: $$u(\bf ...
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Existence of conjugate harmonic function.

In our class we are looking at the following theorem. Theorem (and definition). For any harmonic function $u: U \to \mathbb{R}$ on a star-shaped, or simply connected, or diffeomorphic to $\mathbb{R}...
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Trouble with Poisson integral

I'm continuing my studies about the space $\mathbb{T}$ and I reach the point in which are introced the Harmonic functions. Well up to now I have a little trouble with understanding the Poisson's ...
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27 views

What is the Laplace form of the pendulum equation derived from the equation of motion of a pendulum?

Simple question really. $L (d^2/dt^2) \theta = - g \cdot \theta $ Or $ I \theta \ddot (t) + m \cdot g \cdot l \cdot sin ( \theta \cdot t ) $ Assuming $ sin \theta = \theta $ for angles less than ...
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Logarithm of the norm of a Banach space-valued holomorphic function is subharmonic?

Given a holomorphic function $f$ in some open subset $G\subset\mathbb{C}$, it is well-known that the real-valued function $z\in G\mapsto\log|f(z)|\in\mathbb{R}$ is a \emph{subharmonic} function, i.e., ...
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In Laplace's equation, why is it that there is only one solution for a particular boundary value?

I know that there is a unique solution to Laplace's equation that has particular boundary value. But I do not understand why this is the case. Thanks for your help in advance! PS: this is a follow-up ...
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Multipole expansion of solution to the Poisson equation

In electrodynamics I have seen the following: Let $\phi$ be a solution to the Poisson equation $-\Delta \phi= \rho$, and assume that $\rho$ is compactly supported. Then we can expand $\phi$ as the ...
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1answer
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Why do we need to know the value of a function at the boundary to determine that there is one, unique solution to Laplace's equation?

I'd like to draw your attention to the First Uniqueness Theorem in context of Laplace's equation. It states that if we know the value of a function $V$ at surface of a volume, then the solution to ...
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45 views

Is this function identically zero??

Let D be a bounded domain in R^n. Show that the problem ∆v−v^3 =0 in D; v=0 on ∂D has no solution other than v ≡ 0. I am thinking I need to prove it to be harmonic in order to apply Green's or ...
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1answer
33 views

Existence of a Green's function equivalent to existence of bounded harmonic function

I want to show the following are equivalent on a Riemann surface $W$: Green's function $g_W (p, q)$ exists. There exists a bounded harmonic function on $W$. There exists a positive harmonic function ...
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Prove the existence of cut off function

I want to prove the existence of cut off function: "a smooth "cut-off" function $\Gamma$ such that $0\leq \Gamma(x) \leq 1$, $\Gamma(x)=1$ for $x \in B_{R}(0)$ and $\Gamma(x)=0$ for $x \notin B_{2R}(0)...
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Are all odd harmonic functions are half-wave symmteric?

I know that the Fourier series expansion of a half-wave symmetric function contains only odd harmonics. Is the reverse true?
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Maximal operator inequality $P^*$

The maximal operator function $P^*$ is defined in this way: DEF: If $f\in L^{p}(\mathbb{T})$, $P^*f(x)=sup_{0<r<1}|P_{r}*f(x)|$.Where $P_{r}(t)=\sum_{I=-\infty}^{\infty}r^{|i|}\phi_{i}(t)$ is ...
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1answer
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Brownian mottion and hitting probabilities

Let $D$ be a domain in $\mathbb{R}^d$ and $A$ a measurable subset of its boundary $\partial D$. For $x \in D$ define $$\phi (x) = \mathbb{P}(X_T\in A) $$ where $(X_t)$ is a Brownian Motion in $\...
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Construction of purely radial harmonic function

So my question is In a spherical shell with $1<r<2$ construct a purely radial harmonic function v such that it takes the values $5$ and $4$ at $r=1$ and $r=2$ , respectively I know that I ...
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Smoothness of $C^2$ harmonic functions on Riemannian manifolds

Let $(M,g)$ be a smooth Riemannian manifold. I've seen mentioned before that a $C^2$ harmonic function $f:M\to \mathbb{R}$ (that is, such that $\Delta f=0$) is automatically $C^{\infty}$. I've seen ...
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Question related to PDE, construction of a purely radial harmonic function (need help)

In a spherical shell with $1 < r < 2$ construct a purely radial harmonic function $u$ such that it takes the values $5$ and $4$ at $r = 1$ and $r = 2$ , respectively I know I should start using ...
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Vortex solution of Laplace equation (XY model)

The hamiltonian of XY model, which is closely connected with BKT - transition is following: \begin{equation} H=\frac{J}{2} \int \text{d}^2 r \, \nabla \varphi \cdot \nabla \varphi, \quad \...
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1answer
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Prove that $v(x)=\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ is harmonic

Let $\Omega$ be an open set in $\mathbb{R}^n$ ($n\geq 3$), and $u:\Omega \to \mathbb{R}$ be a harmonic function. Let $$\Omega'=\Big \{x:\frac{x}{|x|^2}\in \Omega \Big \} \mathrm{\ \ and\ \ }v(x)=\...
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1answer
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dimension of a vector space of harmonic functions among polynomials in two variables

I'm searching for harmonic functions inside a set of HOMOGENIOUS polynomials in two variables. Let's say that $$P_n = \{\sum_{i+j = n} a_{ij} x^i y^j \quad|\quad a_{ij} \in \mathbb{R}\}$$ Let's write $...
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Why is the p-Laplace Equation quasi-linear?

I'm having trouble seeing why the $p$-Laplace PDE $$\text{div}(|Du|^{p-2}Du) = 0$$ is quasi-linear. Any ideas? For completeness \begin{align} \text{div}(u) &:= \sum_{i=1}^n \frac{\partial ...
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1answer
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Proving a harmonic polynomial in $x$ and $y$ is a linear combination of $\Re(x+\mathrm iy)^n$ and $\Im(x+\mathrm iy)^n$

Prove that a harmonic polynomial in $x$ and $y$ is a linear combination of $\Re(x+\mathrm iy)^n$ and $\Im(x+\mathrm iy)^n$. My train of thought is as follows: Prove that a harmonic polynomial is a ...
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Laplace's equation on Lorentzian manifold

In general relativity one wants to find "harmonic" functions $u$ on (a neighborhood $U$ in) a Lorentzian manifold $(M, g)$. In arbitrary coordinates $\{x^\mu\}$ the equation $\Delta_g u = 0$ reads $$...
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1answer
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Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?

Is there an analytic continuation of the generalised harmonic number $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$ to the positive reals x, for $k>1$? I can’t find anything useful through Google, just ...
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Counter-example to requiring $\Omega$ to be simply connected for existence of harmonic conjugate

I have read the proof that if $u:D\to\mathbb{R}$ is harmonic and $D$ is simply connected then there exists a harmonic conjugate for $u$. I can see why simple-connectedness is required in the proof, ...
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Harnack's inequality for gradient

How to prove the following: If u is a function harmonic and positive in the disk |z| < R, then $|∇u(0)| ≤ 2u(0)/R$? Any hint is welcome. Thanks in advance.
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Subharmonic functions and monotonically increasing integrals

Let $U \subset \mathbb{C}$ be open and $f : U \to \mathbb{R}\cup\{ - \infty\}$ be a subharmonic function. Define $\varphi(r) := \frac{1}{2\pi} \int_{0}^{2\pi} f(a + re^{i\theta}) \ d\theta$. Show that ...
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Dirichlet problem for Laplace equation with square domain

I'm studying for PDE qualifying exams and came across a problem that is giving me issues. The problem is: Given the square $\Omega=\{|x|<1,|y|<1\}$, let $u\in C^2(\Omega)\cap C^1(\overline{\...
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1answer
24 views

Why is $\nabla u (r\cdot 0) = r\nabla u (0)$ true for a harmonic function $u$?

I've stumbled across an older post here trying to solve the same problem the asker of the post had. The solution that was provided stated that for a harmonic function $u$ on $\mathbb{R}^n$ we have ...
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34 views

Try to understand the prove

I am looking at this prove: https://math.stackexchange.com/a/61101/774621 But I don't get how they did this transformation: $u(x)=\frac{1}{\Omega_nr^n}\int_{B_r(x)}u(v)\;\mathrm{d}v=\frac{1}{\int_0^...
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1answer
52 views

Why is it in $C_c^\infty(\mathbb{R}^n)$?

I try to understand the following prove of an old problem: https://math.stackexchange.com/a/61101/774621 There the following function is define: $\displaystyle\phi_r(s)=\left\{\begin{array}{cl}e^{s^...
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1answer
28 views

A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
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1answer
41 views

Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$? [duplicate]

Can you help me with that: Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$ ? I saw this in Signal Processing course and I can’t understand why this is true. Reference: https://dsp....
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1answer
20 views

Computing the Laplacian of a function

This is a function that I have spent a lot of time on, attacking it from several directions, to no fruition I really need some help with this one, the problem is: Prove that: ($\frac{\partial^{2}}{\...

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