Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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30 views

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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75 views

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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3answers
135 views

$Log(z^2)$ 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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43 views

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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25 views

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
20 views

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
110 views

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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87 views

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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15 views

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
39 views

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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79 views

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
84 views

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
70 views

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
31 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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18 views

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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119 views

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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3answers
146 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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41 views

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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57 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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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(\...
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28 views

Maximize derivative of harmonic function on the unit disk

I am interested in maximizing the partial derivative $$\frac{\partial u}{\partial x}(0,0)$$ for $u \colon \mathbb{D} \to [0,1]$ a harmonic function defined on the unit disk $\mathbb{D} = \{(x,y) \in \...
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2answers
90 views

Harmonic conjugate and resulting analytic function

I can't quite figure out the last part of this problem... Find a harmonic conjugate of the harmonic function $u(x,y)= x^3-3xy^2$. Write the resulting analytic function in terms of the complex ...
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16 views

Subharmonic function bounded by linear functions

Problem: Let $\Omega = \{(x,y) \in \mathbb{R}^2 : xy > 0\}$ and suppose $u \colon \overline{\Omega} \to \mathbb{R}$ is continuous and subharmonic on $\Omega$ and satisfies $$u(x,y) \leq \sqrt{x^2 + ...
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53 views

Laplace Equation in Rectangle: Boundary Value Problem [closed]

Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(...
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12 views

Max vs. min bound for nonnegative harmonic function

Problem: Let $\Omega$ be an open, bounded, simply connected subset of $\mathbb{C}$ and let $u \colon \Omega \to \mathbb{R}$ be a nonnegative harmonic function. Show that for each compact subset $K \...
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1answer
54 views

Rado–Kneser–Choquet Theorem proof

I was reading the proof of the Rado–Kneser–Choquet Theorem. The statement is there in the image (taken from the book "Harmonic Mapping in the Plane, Duren page-$30$": In the proof, he shows that ...
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71 views

Show explicitly that the gradient of two harmonic functions are orthogonal

So form a prior problem stating that $$u(x,y)=\sin x\cosh y\qquad (1)$$ I found the harmonic conjugate, analytic function and found out that it indeed was harmonic $(u(x,y))$ that is. $$f(z)=\sin z+...
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2answers
59 views

Linear harmonic functions

Let $u(x,y)$ be a harmonic function on $\mathbb{R^2}$ and $v(x,y)$ be a harmonic conjugate of $u(x,y)$ on $\mathbb{R^2}$. Suppose that the partial derivative $v_x(x,y)<C$ for a real constant $C$. ...
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1answer
58 views

Laplace Equation with Inhomogeneous Boundary Condition

I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous. I'm having trouble deciding how to handle one of the boundary ...
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1answer
49 views

2D partial differential equation with boundary values (Dirichlet problem)

I'm trying to solve a differential equation (Dirichlet problem): $\Delta \varphi(x, y) = 0$ $0 \leq x \leq a, \ \ \ 0 \leq y \leq a$. The boundary values are: $\varphi(a, y) = \varphi(x, a) = \...
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21 views

harmonic function degenerating at infinity

If $\Omega\subset R^n$ is a bounded domain with smooth boundary. Prove that there is a positive harmonic function $u$ on the complement of $\overline{\Omega}$, with $u=1 $on $\partial\Omega$ and ...
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1answer
29 views

How upper bound of harmonic sum is proven

I know it’s that kind of $1 + \log_{2}{n}$, I don’t know where did that come from etc...
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86 views

Partial Differential Equations: Dirichlet problem for the Laplace equation on half-space with boundary condition

Consider the Dirichlet problem for the Laplace equation on half-space with the boundary condition $f(x)=\chi _{[0,\infty)}(x)$. Use the exact solution and find a functon $g:\mathbb{R}\rightarrow \...
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1answer
80 views

Nonconstant solutions of homogeneous Poisson equation on domain without boundary

This answer on physics stack exchange says For a compact domain without boundary (such as the surface of a sphere), you don't need any boundary conditions: there are no non-constant harmonic ...
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Partial Differential Equations: Consider the Dirichlet problem for Laplace equation on half-space with B.C, use exact solution, find a function g

i have no idea how to approach this problem! any help would be highly appreciated! thank you
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97 views

2D Laplace Equation in Polar Coordinates on Half-Disk with Neumann Conditions

Assume the 2D Laplace Equation $$\nabla^2 u =0 $$ in polar coordinates $(r, \theta)$. The general solution is $$u(r,\theta) = \frac{A_0}{2} + \sum^{\infty}_{n=1}r^n[A_n\cos(n\theta) + B_n\sin(n\theta)...
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1answer
79 views

Green's function for $\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}$

Compute the Green's function for the Laplacian, for the region $$\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}.$$ My approach is to use a reflection argument similar to the one used for ...
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92 views

Uniqueness of the potential flow past a cylinder

I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \...
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21 views

Relation between Warping Function and Solution of Poisson's Equation

In the solution of torsion problem for non-circular cross-sections, the warping function is defined. Without going into the details, the torsion constant is defined as follows: $$ J = \int_A \left[ \...
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27 views

Solve this PDE using Fourier Transform

Question: Solve the following system using Fourier Transform: \begin{alignat}{2} & \nabla^2 f = 0 & z<0 \\ & \frac{\partial f}{\partial z} = \frac{\partial g}{\partial x} \quad \frac{\...
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If v is harmonic conjugate of u, then the harmonic conjugate of $3u^2 − v^3$ is harmonic conjugate of

Attempt: I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to $3u^2 − v^3$ using any other holomorphic functions,...
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1answer
106 views

Does having harmonic functions $u(x,y)$ and $v(x,y)$ guarantee having an analytic function?

Verify that each given function $u$ is harmonic (in the region where it is defined) and then find a harmonic conjugate of $u$. (a) $u=y$ I was able to verify that $u$ is harmonic pretty easily. But ...
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1answer
260 views

Finding the harmonic conjugate of a function

The problem: Find the harmonic conjugate of $G(x,y)= 2v^2(x,y)-2u^2(x,y)$ My attempt to solving it I know that "If two given functions u and v are harmonic in a domain D and their first-order ...
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1answer
58 views

Find a and b values so a given function is harmonic

The Problem: Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ be an entire function. If $g(z)=au^2(x,y) - bv^2(x,y)$ find values for a and b so $g(z)$ is a harmonic function. My attempt to find a solution: Well ...
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29 views

Show that the equation defines a harmonic conjugate

Suppose $u$ is a twice continuously differentiable real-valued harmonic function on a disk $D(z_0;r)$ centered at $z_0 = x_0 +iy_0$. For $(x_1, y_1) \in D(z_0;r)$, show that the equation \begin{align*...
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55 views

An estimate for the conjugate Poisson Integral on the unit disk.

Let $f \in L^1(\Bbb{T})$ and $Q_r(\theta)=\frac{2r\sin{\theta}}{1-2r \cos{\theta}+r^2},r \in[0,1]$ the conjugate Poisson kernels and $P_r(\theta)=\frac{1-r^2}{1-2r \cos{\theta}+r^2},r \in[0,1]$ and $\...
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41 views

Conditional expectation of a bounded harmonic function

Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map ...
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18 views

find a harmonic function $v$ such that $v$ takes 1 on $L_1$, $3$ on $L_2$, and equals $5$ on $L_3$

Let $D=\{z: \Im z>0 \text{ and } |\Re z|<\pi/2\}$ be a half strip. Let $L_1$ be the left boundary $L_1:=\{z:\Re z=-\pi/2\ \text{ and } \Im z>0 \}$ and $L_2=\{z:\Re z=\pi/2\text{ and } \Im z&...
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21 views

The dirichlet and harmonic functions why they are important

I am wondering why finding a function that is harmonic on the sphere and that respect some conditions on the frontiere of the sphere is important ? This is called the Dirichlet problem, and I don't ...
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1answer
30 views

P-norm of the Poisson Kernel

If we have $P_r(\theta)=\sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}=\frac{1-r^2}{1-2r\cos(\theta)+r^2},$ Is there a quick way to compute the norm $\|P_r\|_p$, or an upper estimate?