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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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0answers
49 views

Bound on gradient positive harmonic function

My question is essentially the same as An inequality concerning an harmonic function , however I did not find the answer given satisfactory. To restate it, I would like to solve the following: Let $h$...
2
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0answers
46 views

Mapping a curve-sided quadrilateral to a rectangle

I am currently investigating different ways of solving the Laplace equation $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial z^2} = 0 $$ numerically on the domain $\Omega$ shown as ...
6
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2answers
462 views

Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
0
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1answer
156 views

Distributional Laplacian of $\log|F(z)|$ Where F is Entire

Let $f(z) = \log|F(z)|$, where $F: \mathbb{C} \rightarrow \mathbb{C}$ is entire. Then $f$ defines a distribution on $\mathbb{R}^2$, and we want to show that its distributional Laplacian is $$\Delta f ...
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0answers
60 views

Generalized Laplace equation

When considering electric potential $\Phi(\vec{r})$ in the presence of dielectric material described by relative permittivity $\epsilon(\vec{r})$, one has to solve the generalized Laplace equation $$\...
6
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1answer
104 views

$L_f(z) = \frac 1 {2 \pi i}\int_{ \mathbb{T} } \frac{ \zeta+z}{ \zeta ( \zeta -z)} f( \zeta ) d\zeta$

I'm trying to prove that for any harmonic function $u$, we have : let $ \Omega \subset \mathbb{R}^2$ and $ \overline B(0,R) \subset \Omega $ $$ u \colon \Omega \to \mathbb R $$ $$\forall z \in ...
9
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3answers
285 views

Heat Equation + Uniform Convergence in time -> Harmonic Limit

Assume we have $u \in C^3(\mathbb{R}^n \times (0,\infty))$ satisfying the heat equation $$ \Delta u(x,t) = \partial_t u(x,t)$$ and a function $u_0:\mathbb{R}^n \to \mathbb{R}$ with unknown regularity (...
0
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0answers
37 views

Is there a pattern for closed and co-closed $n$-forms on $\mathbb{R}^{2n}$?

Consider $\mathbb{R}^{2n}$ with its standard Euclidean Riemannian metric. Let $\omega \in \Omega^n(\mathbb{R}^{2n})$ be an $n$-form. I am trying to understand if there is a succinct way to express ...
0
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1answer
47 views

Compute the limit for a harmonic function given two known limits

This problem is from a set of exercises that I have. It states: Let $u\in C(\overline{\mathbb{R}_+^2})$ be a bounded harmonic function in the upper half plane $\mathbb{R}_+^2$, satisfying $u(x,0) \...
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0answers
82 views

Reference Markov martingale Harmonic function

I've just finished a course of stochastic process (discret martingale and markov chain). I would like to go further, I heard it exists a link between martingale markov process and harmonic functions. ...
4
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1answer
66 views

Non-trivial entire harmonic function in plane

Is it possible to explicitly find a harmonic function $u \in C^2(\mathbb{R}^2)$ such that \begin{equation}\tag{$\dagger$}\label{eq:dag} u(x,1) = u(x,-1) = 0 \end{equation} for all $x \in \mathbb{R}$? ...
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1answer
60 views

Uniqueness of solutions of boundary value problem

I know that $u_1 = ln|x^2+y^2|$ is harmonic. Knowing that $u_2 = 0$ is harmonic, I can see that boundary values for $u_1$ on unit circle coincide with boundary values of $u_2$. My question is does ...
1
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1answer
213 views

A maximum principle for bounded functions in unbounded domain

Let $U \subsetneq \mathbb{R}^2$ be a domain. Suppose that $u \in C^2(U) \cap C(\bar{U})$ is a bounded harmonic function such that $u \leq 0$ on $\partial U$. If $U$ is bounded, then the maximum ...
6
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3answers
275 views

Bounded Harmonic Functions on the Disk

Denote by $\mathbb{D}$ the open unit disk in $\mathbb{R}^2$. Is it possible to find a bounded harmonic function $u : \mathbb{D} \to \mathbb{R}$ that is not uniformly continuous? I tried using ...
0
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1answer
177 views

Wave Equation and Fourier Series

I was given a guitar string of length 1 with fixed endpoints. My $f(x)$ is $2x$ if $(x \le 0.5)$ and $-2x+2$ if $(x \gt 0.5)$. My initial velocity is 0. $f(x)$ is the initial position I was first ...
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0answers
59 views

Finding my own harmonic oscillator (differential equations assignment)

In my intro to differential equations class, we were assigned a project from the textbook in which we are to find out own harmonic oscillator. Here are some details: " In the text, we claim that the ...
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0answers
53 views

Linearity solving Laplace equation in rectangular domain with non-homogeneous conditions

Strauss' book on PDEs explains that to solve Laplace's equation in a 2D rectangular $a < x < b, \ c < y < d$ domain with non-homogeneous conditions $$u(a,y) = f$$ $$u(b,y) = g$$ $$u(x,c) = ...
3
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1answer
116 views

Harmonic functions in the half-plane

Denote by $\mathbb{H}$ the upper half-plane $$ \mathbb{H} := \left\{ x \in \mathbb{R}^n : x_n > 0\right\}. $$ Suppose that $u \in C^2(\mathbb{H}) \cap C(\bar{\mathbb{H}})$ is a bounded harmonic ...
0
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1answer
54 views

For which subsets does the harmonic to analytic connection hold?

I'm a bit confused on the choice of sets that authors choose and why. For example : "Any harmonic function $u$ on an open subset $\Omega$ of $R^2$ is locally the real part of a holomorphic function." ...
0
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1answer
536 views

Analytic functions having harmonic real and imaginary parts.

I've bee set the following question in a homework assignment for my complex analysis class, but have literally no idea what it means by sufficiently regular. Let $f : \mathbb{C} \to \mathbb{C}$ be an ...
0
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0answers
114 views

Mixed Cauchy and Dirichlet and unspecified boundary conditions for Laplace equation on $I^2$

I am looking for a reference where the following problem is discussed: $u \in C^{\infty}(I^2)$ so that $\Delta u = 0$ $u(0,y) = f(y)$, $u(1,y) = g(y)$, $u(x,0) = h(x)$ $\nabla u(x,0) \cdot \hat{n} (...
2
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1answer
266 views

Proving an integral formula containing the Poisson kernel

Specifically, the question is as follows: Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$ and suppose that $f:\bar{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function such that both $\...
1
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1answer
39 views

Why is it that for a hamonic $u$, $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function there?

Let $u$ be a harmonic function on a connected open set. If $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function. This question arises from an answer to this post Please do ...
0
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1answer
93 views

How to find the harmonic conjugate of $v(x,y)=\log((x-1)^2 +(y-2)^2)$?

So I have found out $$\frac{\partial v}{\partial x} = \frac{2x-2}{(x-1)^2 +(y-2)^2},\ \frac{\partial v}{\partial y} = \frac{2y-4}{(x-1)^2 +(y-2)^2}.$$ Using the Cauchy Riemann equations, I find: $$\...
2
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1answer
75 views

Let $v(x) = \int_{B(0,R)} \frac{c}{||x-y||^{n-2}}\mathrm{d}y$. Show that for $x \in B(0,R)^c$ we have $v(x) = c_1||x||^{2-n} + c_0$

Let $$v(x) = \int_{B(0,R)} \frac{c}{||x-y||^{n-2}}\mathrm{d}y$$ Show that for $x \in B(0,R)^c$ we have $$v(x) = c_1||x||^{2-n} + c_0$$ Where $B(0,R) \subset \mathbb{R}^n$ is the ball centered at $0$ ...
0
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0answers
41 views

A function analytic in the unit disk belongs to the class Nevanlinna if and only if it is the quotient of two bounded analytic functions

I'm trying to understand a part of this proof from Duren, in the converse, I don't see it clear when it says "by analytic completion of the Poisson Formula,..." and then the result; I tried to prove ...
1
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1answer
353 views

Harmonic functions and Maximum Modulus Principle. (Real harmonic function implies holomorphic function?)

Theorem. (Maximum Modulus Principle) $\;$If $f$ is a non-constant holomorphic function in a region $\Omega$, then $f$ cannot attains a maximum in $\Omega$. Problem. Prove the maximum principle ...
3
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2answers
162 views

PDE Laplace equation. Integral representation form and Green function

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and assume that for any $y \in \Omega$ there is a function $h_{y} \in C^{2}(\overline{\Omega})$ such that \begin{equation} \label{eq8.1} \begin{cases} ...
0
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2answers
214 views

The harmonic conjugate of the function

Let $u + iv$ be analytic, and $u(x, y) = \cosh{(x)}\cos{(y)}$. Find the harmonic conjugate function $v(x, y)$. The harmonic conjugate function is given by $ \begin{align} v(z) &= \int_{z_0}^z ...
1
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1answer
103 views

Interior gradient estimate harmonic function: decay estimate

I found this problem Proposition but I am completely stuck. Let $u$ be an harmonic function satisfying $$ \int_{B_1(0)}|\nabla u|^2 \mathrm{d}x \leq 1, $$ where $B_1(0)$ is the unitary ball in $\...
2
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1answer
32 views

counter-example to holomorphic transformations of harmonic functions?

I've been taught in class that if $\phi(x,y)$ is a harmonic function and $f(z)$ is a holomorphic function viewed as a function $\mathbb{R}^2\rightarrow\mathbb{R}^2$ then $\phi \circ f$ is also a ...
0
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0answers
65 views

Laplace's Equation - Existence and Uniqueness with Robin conditions on an annulus

Let $k$ be a non-zero real number. Consider the problem $$ \nabla^2 \phi = 0 \ \ \ \mbox{for} \ \ \ 1 \leq r \leq 2, \ \ \ \ \alpha\phi + \frac{\partial\phi}{\partial r} = k\cos\theta \ \ \ \mbox{...
2
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0answers
65 views

Harmonic functions interpolation

Denote by $B = \{(x,y) \in \mathbb{R}^2|x^2 + y^2 < 1\}$ the open unit ball in $\mathbb{R}^2$, and by $S$ it's boundary, i.e. the unit sphere. For some $n > 0$ let $x_1,...,x_n \in B$, $y_1,...,...
2
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0answers
54 views

Correct form of the Laplacian on a 1-D ellipse, and it's solutions

I wanted to derive the Laplacian operator for a 1-D ellipse, and it seemed to me that there are two equivalent approaches: 1) Start with 2-D elliptic coordinates $$ x = a \cosh(\mu) \cos(\nu)$$ $$ y ...
1
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0answers
160 views

Mapping the solution to Laplace equation with Dirichlet and Neumann boundary conditions from half-plane to quadrant

Background: I have obtained the solution to the Laplace equation in the upper half-plane $(y \geq 0 , -\infty < x < \infty)$ with a function value of zero and prescribed normal derivative of $\...
1
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1answer
49 views

Hardy space and extension to L2: convergence theorems

Let $f \in H^{2}(\mathbb D)$. I'd like to show that $ f\in L^{2}(\mathbb T)$, with: $\Vert f \Vert_{L^{2}(\mathbb T)} \le \Vert f \Vert_{H^{2}(\mathbb D)}$, maybe up to a constant. $f \in H^{2}(\...
5
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1answer
58 views

Confusion with the formula for harmonic conjugate

According to here, the harmonic conjugate of a harmonic function $u$ is given by $$v(z)=\int_{z_0}^z u_xdy-u_ydx+C$$ where $C$ is a constant, while in here, the harmonic conjugate is given by $$v(z)=...
2
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0answers
65 views

Harmonic functions on an irreducible recurrent Markov chain are constant

I would like to show that if $(X_n)$ is an irreducible recurrent Markov chain (on a countable space), and $f \geq 0$ is harmonic, then it is constant. I do the following: for any $x$, $f(X_n)$ is a ...
1
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0answers
66 views

Find harmonic function vanishing on the boundary and with a specific bound

Find all harmonic functions $u$ in a half-plane $H$ so that $u=0$ on $\partial H$ and $\vert u(x)\vert \le \vert x\vert$ in $H$. This domain is not bounded. If it's bounded, then after using the ...
1
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0answers
21 views

An (sort of) inverse Dirichlet problem

Let $G = (V, E)$ denote the $n \times n$ integer grid, with the natural boundary $\partial V$. If $f$ is any real valued function defined on $\partial V$, then it is well known that $f$ can be ...
0
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1answer
42 views

Finite invariant measures for reversible Markov chain

I look at a reversible Markov chain on a countable set $G$, i.e. if $p_{xy}$ is the transition probability from $x$ to $y$, there is a positive function $\pi$ such that $$ \pi(x) p_{xy} = \pi(y) p_{...
1
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0answers
46 views

Surface Integral - Mean Value Property

I have been given this question to solve however I'm having some difficulty solving it as I am quite new to Partial Differential Equations: Let $ a = (2,2) $ and $ r =5 $ Compute the following ...
1
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0answers
70 views

Difference of fundamental solutions

I am reading a set of notes in which the author claims that the difference between any two fundamental solutions (to the Laplacian) is harmonic. That is, if $E_1$ and $E_2$ are fundamental solutions ...
1
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1answer
43 views

Considering two regions and a holomorphic function, show the following

I'm very lost on how to do this question. Do I use cauchy-reimann equations somehow? Thank you for your help!
-1
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1answer
176 views

partial differential equations, Laplace equation, boundary conditions [closed]

For the stated problem I got the following equations after applying separation of variables, I am confused as to how to apply the mentioned boundary conditions to this problem : \begin{align} G''(x)+(...
0
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2answers
70 views

Find a harmonic function u that vanishes on C[0, 2] and satisfies $u(3e^{i\theta}) = 65\cos(2\theta)$

Consider the annulus $G = \{2 \leq |z| \leq 3 \}$ (this is basically a ring with inner radius $2$ and outer radius $3$). Find a harmonic function which vanishes on $C[0, 2] $(this is a circle with ...
5
votes
0answers
59 views

What is the closure of $h^2(U)$ in the topology of the uniform convergence on upper half planes?

Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$. Let $h(U,\mathbb{R})$ be the set of real harmonic functions defined on $U$. Define $$h_0(U,\mathbb{R}):=\left\{u\in h(U,\mathbb{R})\ |\ \left(...
0
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2answers
49 views

PDE Laplace equation Calculate $u(x,0)$.

Let $u(x,y)$ be a harmonic function in the domain $0\le x^2 +y^2\le 1$ that satisfy the following terms at the rim of the circle: $$\begin{cases} u(x,y)=3y & \text{if y>0} \\ u(x,y)=0 &\...
3
votes
1answer
113 views

In general, what information is given by $\nabla^2 f \geq 0$?

Since there are many directions one can take when studying this equation, I am curious: Given a function $f \in C^2$ defined on some open set, what information is given by $\nabla^2 f \geq 0$? Please ...
1
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0answers
50 views

Equivalence of local mean value property and mean value property

Let $\Omega \subseteq \mathbb{R}^n$ open and connected. Let $u$ be continuous. We say $u$ satifies the Mean-Value Property (MVP) if $\forall x \in \Omega, B_{\epsilon}(x) \subseteq \Omega$, we have ...