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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Exercise Jurgen Jost's PDE show that u harmonic and nonnegative is constant

2.5: Let u be harmonic and nonnegative, show that u is constant. (Hint use the previous exercise). The previous exercise was posted in another question, stated the following. 2.4: Let $u:B(0,R)\...
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Poisson problem, minimun value of subharmonic function in the interior.

Consider the problem: $$ -\Delta u = - f(x_1,x_2), \text{in } \Omega$$ $$ u = f(x_1,x_2), \text{in } \partial\Omega$$ Where $\Omega=B(0;1) \subset \mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$. Is it ...
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Exercise in Jurgen Jost: Show that if $u$ is harmonic then $v$ is harmonic

Let $\Omega \subset \mathbb{R^3-\{0\}}$ and $u:\Omega \rightarrow \mathbb{R}$ harmonic in $\Omega$. Show that $$v(x^1,x^2,x^3):=\dfrac{1}{|x|}u\Big(\dfrac{x^1}{|x|^2},\dfrac{x^2}{|x|^2},\dfrac{x^3}{|x|...
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Finding / creating harmonic functions with given criteria

Find a function $u$ harmonic on $\{Im(z)>0, Re(z)>0\}$ with boundary values $0$ on $\{Im(z)>0, Re(z)=0\}$ and 1 on $\{Im(z) = 0, Re(z) > 0 \}$. How does one go about this? My professor ...
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Generalize $\sum_{n=1}^{\infty}\frac{H_n^2-H_n^{(2)}}{(n+1)(n+2)}=2$

In this paper on section [5], Recently J. Choi [4, Corollary 3] proved a sequence of identities: $$\sum_{n=1}^{\infty}\frac{H_n^2-H_n^{(2)}}{(n+1)(n+2)}=2\tag1$$ Let just generalize $(1)$ $$\sum_{...
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2answers
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finding Ker(T) of a parameter's linear transformation

I am suppose to find the ker(T) of linear transformation of: $$ G\begin{pmatrix}a & d \\ c & b\end{pmatrix}= a+\frac{b+c}{2}x+\frac{b-c}{2}x^2 $$ the form $T:V \to W$ My problem is that I ...
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Regularity for weak solution of Poisson problem in a rectangle

Let $\Omega=(0,1)^2$. Let $u$ be a weak solution of $\Delta u=f$ con $f \in L^2(\Omega)$ e $u \in H^1_0(\Omega)$. I would like to prove that $u \in H^2(\Omega)$. I know that $u \in H^2_{loc}(\Omega)$ ...
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Excercise 2.4 in Jurgen Jost's PDE “Harnack's inequality” for harmonic functions defined on a ball

Let $u:B(0,R)\subset \mathbb{R^d}\rightarrow\mathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$\dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)\leq u(x)\leq \...
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Use of Holder inequality in gradient estimate for harmonic function.

While reading the book "Elliptic Partial Differential Equations" by Han and Lin, I failed to understand the proof of the interior gradient estimate for harmonic functions. The theorem says that if $u$ ...
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Is $\ln|z|$ harmonic in the punctured disk [closed]

1. how can i show that $\ln|z|$ is harmonic in punctured disk ? also $\ln|z|$ has no harmonic conjugate in $\Bbb C\setminus\{0 \}$ but has in $\Bbb C\setminus[0, \infty)$.
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Solving the Dirichlet Problem for an infinite strip

I have been looking into the Dirichlet problem and conformal mappings, but am unsure as to how to find a solution $u(x, y)$ for the Dirichlet problem given this information: The region is $U = \{\ x+...
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Are the singular points of harmonic function on the disk always of measure zero?

Let $f : \mathbb D^2 \to \mathbb R$ be a smooth function with no singular points, i.e. $df \neq 0$ on $\mathbb D^2$. (Here $\mathbb D^2$ is the closed unit disk in $\mathbb R^2$). Let $\omega:\mathbb ...
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Liouville's theorem for harmonic functions

I was reading the proof of Liouville's theorem for harmonic functions (in $\mathbb{R}^n$) in Wikipedia, but I could not understand where do they use in that proof the assumption that $f$ is bounded. ...
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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 $h$...
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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 ...
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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. ...
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1answer
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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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Harmonic functions with a boundary condition.

I am looking for a harmonic function. Let $H(x)=x^2$ and let $D=\{(x,z) \in \mathbb{R} \times \mathbb{R}^2 \mid x>1, |z|<H(x)\}$. Here, $|\cdot|$ denotes the $2$-dim Euclid norm. $D$ is an ...
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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 $$\...
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$L_f(z) = \frac 1 {2 \pi i}\int_{ \mathbb{T} } \frac{ \zeta+z}{ \zeta ( \zeta -z)} f( \zeta ) d\zeta$

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 ...
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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 (...
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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 ...
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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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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. ...
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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
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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 ...
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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 ...
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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 ...
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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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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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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) = ...
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1answer
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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 ...
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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." ...
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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 ...
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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} (...
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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 $\...
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1answer
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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 ...
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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: $$\...
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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 $0$ ...
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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 ...
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Is this PDE (Poisson/Laplace equation) well-posed considering I have a very degenerate domain (picture included)?

In case (1) in the following picture we have the standard interior Poisson equation in 2D with Neumann boundary conditions on some smooth domain $\Omega$, subject to a point source at position $y$. I ...
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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 ...
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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} ...
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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 ...
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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 $\...
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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 ...
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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{...
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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,...,...
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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 ...
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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 $\...