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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Gradient estimate and divergence theorem in an open ball (PDE)

I need to prove the estimate $|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$ where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$ I found this ...
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Perron's Method with Measurable Boundary Data

Perron's method is a standard construction to solve the Dirichlet Problem with continuous boundary data on domains with sufficient regularity. My question is about Boundary data that is no longer ...
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converting a function to a harmonic one in dirichlet boundary condition

I am trying to convert to a harmonic one in a manner that the harmonic function to be equal to the original function at $x=0$. is there any method to use for this purpose? thanks
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Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
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Understanding proof: if $u: \Omega \subset \mathbb{R}^N \to \mathbb{R}$ is harmonic and $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant.

I'm trying to solve the following question: If $u$ is harmonic in $\mathbb{R}^N$ e $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant. I've found this solution, which I posted below, but I ...
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Prove harmonic function inequality with the Mean-Value Property

Let $\Omega \subset \mathbb{R}^n$ open and let $u$ be a harmonic function in $\Omega.$ If $K \subset \Omega$ is compact, then prove $$ \sup\limits_{x \in K} |u(x)| \le \frac{n}{\omega_n ~dist(K,...
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limit involving harmonic function

Let $u$ harmonic function in $\mathbb{R}^3 -\{0\}$. I know that $$\lim_{x\to0} \sqrt{|x|} \cdot u(x)=k< \infty$$ I'm trying to show that $k=0$. I tried by contradiction, but I failed and I'm ...
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Harmonic extension for harmonic function

Studying the mean spherical mean and the volumetric mean, this question has occurred to me. The volumetric mean is defined as follows: Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \to \...
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1answer
56 views

Estimate for the fundamental solution of Laplace equation from Evans

After the definition of the fundamental solution of Laplace equation (page 22) $\Phi(x) = \begin{cases} -\frac{1}{2\pi} \, \log(|x|), \, & n=2, \\ \frac{1}{n \, (n-2) \, \omega_n} \, \frac{1}{|x|...
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If partial derivatives of a harmonic function are constant, is the function linear?

Let $u: \mathbb{R}^2 \to \mathbb{R}$ be a harmonic function. If $\frac{\partial u(x,y)}{\partial x} = k_1$ and $\frac{\partial u(x,y)}{\partial y} = k_2,\forall (x,y) \in \mathbb{R}^2, k_1, k_2 \in \...
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Showing that a harmonic function is 0

Let u(x) a harmonic function in $\mathbb{R}^n$ such as: \begin{equation} \int_{\mathbb{R}^n}|u(x)|dx =K< \infty \end{equation} Show thtat $u(x)=0$, $\forall x \in \mathbb{R}^n$. Using the ...
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The Laplacian is pointwise defined on a subharmonic function with bounded (distributional) Laplacian

Let $\phi$ a subharmonic function with (distributional) Laplacian uniformly bounded ie: there are $A, B$ positive real numbers such that $A < \Delta \phi < B$. For some reason, the author of ...
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example of a harmonic function that does not have a harmonic conjugate in non-simply-connected domain [closed]

Theorem 1. Let $S \subset \Bbb R^2$ be an open subset. Every harmonic function $u : S \to \Bbb R$ has a harmonic conjugate if and only if $S$ is simply connected. I would like to do an exercise that ...
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Showing that Poisson kernel for the unit disc is harmonic.

Let $r \in [0,1)$ and $\theta \in [-\pi,\pi]$ and define, $$P_r(\theta) = \frac{1-r^2}{2\pi(1+r^2-2r\cos(\theta))} = \frac{1}{2\pi}\sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}.$$ Then I want to show ...
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1answer
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PDE, Dirichlet problem for a circle

When I'm learning laplace's equation with the case is dirichlet problem for a circle, i know that naturally i have to separate variables in polar coordinate. And i derive from rectangle to polar ...
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1answer
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laplace equation on a triangle

I was recently studying Laplace's equation and set myself the problem to solve it on a isoceles, right triangular domain. Two possible approaches came to my mind: a) transform the coordinates into a ...
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1answer
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Necessary and Sufficient Conditions for the Solution of Neumann Problem

Let $\Omega$ be a domain with smooth boundary $\partial \Omega$, $\bar{\Omega}$ is compact. The Neumann problem:$f \in C^{\infty}(\bar{\Omega})$,$g\in C^{\infty}(\partial \bar{\Omega})$. Find a ...
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Equation for the normal derivative of the solution of Poisson's equation

The solution of Poisson's equation \begin{align*} \Delta u &= 0 ~~~\text{in } B_r \\ u &= g~~~\text{in } \partial B_r, \end{align*} is well known. For some $a\in\Bbb R\setminus \{0\}$ we ...
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Poisson integral on $\mathbb{D}$ is surjective

If $\phi:\mathbb{S}^1\to \mathbb{S}^1$ is a homeomorphism then its harmonic extension on $\mathbb{D}$ is given by, $$ f(z)=\dfrac{1}{2\pi}\int_0^{2\pi} \dfrac{1-|z|^2}{|e^{it}-z|^2}\phi(e^{it}) .$$ ...
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Basis Functions For a 3D Source-Free Field

I have a 3D source-free domain, which is governed by the Laplace equation, $\Delta u = 0$. The field represents an electrostatic field and is measured at a point cloud consisting of $n$ non-boundary ...
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Composition of subharmonic with holomorphic is subharmonic

I need to prove the following claim: PROBLEM: Asuume $U_1,U_2\subseteq\mathbb C$ are domains in $\mathbb C$. Show that if $f:U_1→U_2$ is holomorphic and $u:U_2\to\mathbb R$ is subharmonic and ...
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Forced Harmonic Osscilators

I have to compute the general solution and compute the solution given an initial value. $ y'' + 3y' + 2y = t^2 $ $ y(0) = y'(0) = 0$ I understand the first thing I need to do is find the ...
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3answers
57 views

Complex derivative of complex conjugate is zero

I'm reading a proof for the statement: let $G\subset\mathbb{C}$ be an open set containing all inner regions of piecewise horizontal or vertical closed linear curves in $G$, and let $u$ be harmonic on $...
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Looking for a class name and a potential general solution

Is there a standard class name if any of the following differential equation there similar to Laplace equation and equivalent solution based on legendre polynomial, or spherical harmonics? $\partial^...
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1answer
67 views

Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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2answers
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Is a harmonic function in $\mathbb{R}^2$ which is $o(\ln |x|)$ a constant?

Let $f : \mathbb{R}^{ 2 } \rightarrow \mathbb{R} $ be a harmonic function. Suppose $$\lim _ { | x | \rightarrow \infty } \frac { | f ( x ) | } { \ln | x | } = 0$$ Prove or disprove that $f$ is a ...
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Regularity of harmonic functions

I have a question on a fundamental property of harmonic functions. Let $\Omega \subset \mathbb{R}^d$ be a domain. We define $H^{1}(\Omega)$ by \begin{align*} H^{1}(\Omega)=\{u \in L^{2}(\Omega,m) \...
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separability of the space of harmonic functions on a domain

I tried hard to find a reference for this but I could not; please help me. Let D be an open set in $R^d$ (connected, if necessary), and let H be the set of harmonic functions on D. We may define a ...
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Harmonic function theory, Kelvin Transform

I have the following problem with exercise 5 in the book Harmonic funtion theory by Sheldon Axler, Paul Bourdon and Wade Ramey: Show that if $ n > 2$, then the only harmonic function on $\mathbb{R}...
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Poisson formula on a disc is continuous on the closure

It is written that the function $u$ is harmonic in $\mathbb{D}$ and continuous in $\overline{\mathbb{D}}$ and $u(e^{it})=\phi(t).$ I am unable to prove the continuity. z The continuity in the inside ...
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Can a decreasing sequence of subharmonic functions converge to a discontinuous function?

If $u_n:\Bbb C\to \Bbb R$ are subharmonic, bounded, continuous, and $u_{n+1}\le u_n$ for all $n$, can their limit be discontinuous? By boundedness, it is easy to show that the limit function $u$ will ...
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$Log(z^2)$ analytic for all of the complex plane except origin.

The question was show $\ln{x^2 + y^2}$ is harmonic in two ways. It was very easy to show by LaPlace's equation, but next I have to show it by showing it is the real part of an analytic function. I am ...
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Regarding sequence of positive Harmonic functions

Let $\{U_n\}_{n\geq 1}$ be a sequence of positive harmonic functions on a domain $\Omega$ and let $z_0\in \Omega$. Suppose that $\lim_{n\longrightarrow \infty}U_n(z_0)=\infty$. How does one show that $...
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Solutions to $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ that only depend on r

Find all the solutions of $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ in three dimensions that depend only on $r = 􏰃x^2 + y^2 + z^2$, the radial variable in polar coordinates. Use the following ...
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1answer
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Maximum Principle for estimate norm

I have two functions. A harmonic funcion $u$ in unit ball $B_{1}\subset \mathbb{R}^{n}$ and a function $h$ defined on $B_{1}$ such that $\Delta h=u$ in $B_{1}$ and $h=u$ in $\partial B_{1}$, the ...
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1answer
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Bound on gradient of Harmonic functions

Let $G\subseteq\mathbb{C}$ be a domain and assume $u:G\to\mathbb{R}$ is a harmonic function such that $|u(z)|\leq M$ for all $z\in G$. Show that $|\nabla u(z)|\leq\frac{2M}{r}$ for $0<r<dist(z,\...
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Estimates on Hessian of Solution to Poisson Equation

Let $\Phi:\mathbb{R}^d\to \mathbb{R}$ be the fundamental solution to the Laplace equation, i.e the unique function $\phi$ such that $\Delta \phi = \delta_0$ in the sense of distributions. The solution ...
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Integral of positive part of harmonic function over circles goes to infinity

I have been thinking about this problem off and on for a couple days now to no avail. If in answering this question you could address the things I've tried so far as "on the right track", "not really ...
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1answer
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Geometric intuition for harmonic conjugate functions

It is known that given a harmonic function $u$ of class $C ^ {2}$ defined in a simply connected subset of $\mathbb{C}$ , we can find a function $v$ also harmonic, such that $f = u + iv$, is a ...
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Harmonic Functions With a Pole at the Origin

I'm trying to solve the following problem: Suppose that $u:\mathbb D \setminus\{0\} \to \mathbb R$ is harmonic and that $\lim_{z\to 0} u(z)=\infty$. Show that $u$ can be written as $$u(z)=\...
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1answer
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If $u(x,y)$ is not a harmonic function, can a harmonic conjugate $v(x,y)$ be found?

If $u(x,y)$ is not a harmonic function, i.e. does not satisfy Laplace equation $$\Delta u(x,y) = \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$ (and perhaps is not continously ...
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1answer
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Properties of functions with $0$ second partial derivatives

I have a $n$-dimensional polynomial that I am evaluating on some domain $\Omega \subset \mathbb{R^n}$ $$ f:\Omega\rightarrow \mathbb{R} $$ where I know that all the second partials are zero $$ \dfrac{...
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1answer
28 views

Harmonic functions - proving a limit

I'm trying to work through the proof of this statement: Suppose $\Omega \in \mathbb{R}^d$ is open with $B_R(x) \in \Omega$ and suppose $u \in C^2(\Omega)$. Define $$\varphi (r):= \int_{\partial ...
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Solution to set of harmonic equations

I have a function \begin{equation}\label{equ:Ac_theta} f(\theta) = A_7sin(7\cdot \theta)+...+A_{17}sin(17\cdot \theta) \end{equation} I am interested in finding the value of second highest maximum ...
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Proof of maximal principle on Laplace Equation involving Poisson integral formula

This question appeared on a past PDE exam I found while studying for my finals: Let $u(r,\theta)$ be solution to the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}...
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$ \lim_{n\to\infty}u_n(x,y)=u(x,y) $ is true for a sequence of harmonic functions satisfying $ \lim_{n\to\infty}\int_B |u_n(x, y)-u(x,y)|dxdy=0 .$

(i)Let $ B=\{ (x, y)\in\mathbb R^2:x^2+y^2<1 \} $, and let $ u(x, y) $ be a harmonic function defined on some open set $ U $ containing the closure of $ B $. Prove that $$ u(0,0)=\frac 1\pi\int_{...
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82 views

If $u(r, \theta)$ is a solution of Laplace’s equation show that $u(\frac{1}{r}, \theta)$ is also a solution.

Suppose that $u(r, θ)$ is a solution of Laplace’s equation. Show that $u(\frac{1}{r}, θ)$ is also a solution. So far, I know that if $u$ satisfies Laplace's equation, then $$\Delta u = u_{rr} + \...
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In a elliptic coordinate system,is there any existing solution for Laplace equation?

In a cylindrical system,Bessel function plays a role. And In a spherical system ,Legendre polynomials works in Laplace equations. Is there any solution for Laplace equation in a elliptic system? ...
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41 views

Subharmonic function on punctured disk

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Suppose $u \colon \overline{\mathbb{D}} \setminus \{0\} \to [0,\infty)$ is continuous and subharmonic on $\mathbb{D} \setminus \{0\}$. Show that ...
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48 views

Quotient space of harmonic functions on punctured plane

Let $a_1,\ldots,a_n$ be $n$ distinct points in $\mathbb{C}$ and let $\Omega := \mathbb{C} \setminus \{a_1,\ldots,a_n\}$. Define $H(\Omega)$ to be the space of harmonic functions on $\Omega$ and $R(\...