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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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7
votes
0answers
442 views

Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
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 ...
6
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1answer
206 views

oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
5
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0answers
50 views

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 ...
5
votes
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)=...
5
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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(...
5
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0answers
106 views

Laplace equation in unit square

I would like to solve the $\triangle u(x,y) = 0$ in the unit square, with periodic BC when $x=0,1$ and Neumann condition when $y=0,1$ $$\partial_y u(x,0) = \begin{cases} A \quad &\text{for } 0\...
5
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0answers
143 views

Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit disk in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a smooth immersion . ($df$ is everywhere invertible). Let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
5
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0answers
72 views

Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
5
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1answer
218 views

Topology of solution to a nonlinear eigenvalue problem

Consider the elliptic PDE: $$-\Delta u= f(x) u. $$ Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial \...
5
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0answers
3k views

Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the Cauchy–...
4
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0answers
58 views

Is there a way to combine solutions of Laplace equation from two different Dirichlet boundary conditions?

Let $\phi_1, \phi_2: \mathbb R^2\to\mathbb R$ be solutions of Laplace equation with Dirichlet boundary: $$ \nabla^2\phi_1 = 0,\quad\quad \phi_1(\mathbf x) = a_1 \quad\mbox{if}\quad\mathbf x\in\Omega_1 ...
4
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0answers
54 views

Regularity for the harmonic equation in coordinates

$\newcommand{\lap}{\Delta}$ $\newcommand{\al}{\alpha}$ $\newcommand{\be}{\beta}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\pl}{\partial}$ Let $(M,g),(N,\eta)$ be smooth ...
4
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0answers
103 views

How to prove this Lambert W integral representation?

I was wondering how to prove this Lambert W identity. $$ - \pi W(-1) = \int_{-\infty}^{\frac{-1}{e}} \Im ( W ' (x) ) \ln(1 + \frac{1}{x} ) dx $$ Maybe with contour integration ??
4
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0answers
91 views

Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
4
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0answers
87 views

Show that the “Hartogs Regularity Radius” $R(z)$ is subharmonic

Exercise I'm a little stuck on an Exercise in Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range. The Exercise (E.II.5.1) is as follows (here $\...
4
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2answers
778 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 u}{\...
4
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0answers
101 views

Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove $u=...
4
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0answers
196 views

is Poisson's kernel always integrable?

Let $E$ be a smooth domain. The Green function $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation and for fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function in ...
4
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0answers
1k views

Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
3
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0answers
24 views

Harmonic circle-valued maps

Let $M$ be a closed Riemannian manifold. A circle-valued function $u : M \to \mathbb{S}^1$ is harmonic if the associated one form $h_u = u^*(d \theta)$ is harmonic in the Hodge sense: $dh_u = 0$ and $...
3
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0answers
47 views

Prove that if $u:\mathbb{R}^d \to \mathbb{R}$ is harmonic with integrable square then $u\equiv 0$.

Prove that if $u:\mathbb{R}^d \to \mathbb{R}$ is harmonic and $$\int_{\mathbb{R}^d}u^2(x)dx=M<\infty$$ then $u\equiv 0$. I'm not sure what the intended method is because there are no solutions,...
3
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0answers
40 views

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 \...
3
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0answers
40 views

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) \...
3
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1answer
34 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 $...
3
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0answers
77 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)=\...
3
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0answers
91 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 \...
3
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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 $$\...
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 ...
3
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0answers
30 views

Approximate indicator of sphere with derivative of test function

Suppose I know that $$ \int_{\Omega} \varphi |Dv|^2 = \int_{\Omega} v\; Dv\cdot D\varphi $$ holds for every $\varphi \in C^1_c(\Omega)$ (this is an easy identity for harmonic functions but that is ...
3
votes
1answer
147 views

Harmonic function with boundary conditions

I need to find a nontrivial function $f:\mathbb{R^2}\setminus\mathbb{D}\rightarrow \mathbb{R}$ ($\mathbb{D}$ denotes the unit disk) such that $\nabla^2f=0, \nabla f $ tends to zero as point $p$ ...
3
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0answers
137 views

Mean value theorem for vector laplacian

It is well known that all solutions of the Laplace equation $\nabla^2 u = 0$ satisfy the mean value theorem: the average value of $u$ over a sphere equals its value at the center of the sphere. My ...
3
votes
0answers
84 views

Estimating the $L^p$ norm of a second derivative of a solution of the Laplace equation

Consider Dirichlet boundary value problem on unit disk : $u_{xx}+u_{yy}=0$. Then, is there a constant $c>0$ satisfying the following? -For every solution $u$ that $\int_0^{2\pi}\left|\frac{d^2u(e^...
3
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0answers
73 views

Calculating Sobolev norm on boundary by extending the map on M as a harmonic map

It is known that the natural trace map $W^{1}(M) \ni \varphi \rightarrow \varphi|_{\partial M} \in W^{1/2}(\partial M)$ is continuous and onto. Since the Dirichlet problem $\Delta \varphi=0$, $\varphi|...
3
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0answers
109 views

Pluriharmonic functions are harmonic on submanifolds?

Let $X$ a Kähler manifold and $f:X\to \mathbb C$ a function. $f$ is said to be pluriharmonic if its restriction to each curve in $X$ is harmonic. Why is then $f$ harmonic? Does this simply follow ...
3
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1answer
72 views

How do I use Laplace's equation to solve this differential equation?

Suppose we have Laplace's equation for some $u(x,y)$ as: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$ for $x \geq 0$, and $0 \leq y \leq a$, and that $u(x,y) \to 0$ ...
3
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0answers
476 views

Using Green's function to solve 2d laplace equation

Consider a domain $D : {(x,y) : x>0 , y>0}$ Let $\mathbf{x}= (x,y)$ and $\mathbf{\xi}= (\xi_x, \xi_y)$. Then the Green's function satisfying $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
3
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0answers
132 views

Closed level sets of a harmonic function

Suppose that $U$ is an open, connected subset of complex plane. Are there any non-constant harmonic functions on $U$ which have at least one closed isocurve? My results so far: If U is simply ...
3
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0answers
52 views

Polar Laplace equation on partially bounded domain - what am I doing wrong?

I am trying to solve the following problem: Let $\Omega$ denote the region $\{ (r, \theta) : r>1, 0 < \theta < \pi \}$. Find the bounded solution to the problem: $\begin{align}\nabla^{...
3
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0answers
84 views

biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
3
votes
1answer
152 views

Dirichlet problem to the ball with boundary data $1-2y^2$.

Let $\omega=\{(x,y):x^2+y^2<1\}$ be the open unit disk in $\mathbb R^2$ with the boundary $\delta\omega$.If $u(x,y)$ be the solution of Dirichlet problem $$\begin{cases} u_{xx}+ u_{yy}=0 & \...
3
votes
0answers
62 views

$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem which I'm not sure what to do. Let's see the hypothesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \...
3
votes
2answers
464 views

All second partial derivatives of harmonic function are $0$

I am given this question as a homework assignment. Assume that $f$ is from $\mathbb R^2$ to $\mathbb R$ and has a strict local maximum at $(x_0, y_0)$. prove that all second partial derivatives of ...
3
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0answers
218 views

How to differentiate a harmonic function presented by Poisson integral formula

Let $h(x+iy)$ be a harmonic function in the open neighbourhood of the closed unit disc $\overline\Delta(0;1)$ of $\mathbb{C}.$ Then it can be presented by Poisson integral formula in the following way:...
3
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0answers
291 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=\Bbb C \setminus \{0, 1\}$. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0 \log |z|−a_1 \log |...
3
votes
0answers
341 views

Prove or disprove: If $h$ is harmonic on $E$, then $h$ is constant on each $C_i$

For a general finite Markov chain $(X_n)_{n\in\mathbb{N}_0}$ with state space $E$ and transition matrix $P=(p_{x,y})_{x,y\in E}$, not necessarily irreducible, we define the linear space of harmonic ...
3
votes
1answer
610 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
3
votes
0answers
49 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
3
votes
1answer
702 views

Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.

Is the form correct for the conjugate harmonic? Attempt: First, we are given \begin{align*} \log \left| z \right| &= u(x,y) + iv(x,y) = \log \sqrt{x^2 + y^2} + i \cdot 0 \\ u(x,y) &= \log \...
3
votes
0answers
138 views

Subharmonic function and holomorphically parametrized integrals

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. Consider the map $N(\lambda)=\log \int |f_\lambda(z)| ...