For questions regarding harmonic functions.

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Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to ...
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35 views

Find all the harmonics functions constants on the rays

I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you. Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the ...
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Non trivial boundaries for laplacian equation on rectangle

1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically? $$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ ...
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1answer
52 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 & ...
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65 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
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1answer
47 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a ...
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147 views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
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60 views

The average of a subharmonic function on a circle increases with radius

Let $u$ be a subharmonic on open set $\Omega$. Let $a\in\Omega,R>0$ such that $B(a,r)\subset \Omega$. Prove $$v(\rho)=\int_0^{2\pi}u(a+\rho e^{it})dt$$ is a monotone increasing function on ...
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3answers
65 views

Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
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36 views

Extending a harmonic function

Suppose that $u$ is a harmonic function on some open set $U$ (assume that $\overline{U}$ is compact). It is well known than in this case $u$ is smooth. Is it true that we can extend $u$ to the whole ...
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How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
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2answers
41 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...
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$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to ...
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Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
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40 views

Normal component of Laplace equation solution

Suppose $V\subset\mathbb{R}^3$ is a bounded region with a smooth boundary and that we are given $f:\partial V\rightarrow\mathbb R$. From classical PDE theory, the solution of the Laplace equation ...
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237 views

Is there a reason why Harmonic functions are defined on open sets?

Whenever I see a definition of a harmonic function, it's always defined as follows A function $f : U \to \Bbb{R}$ is called harmonic (where $U$ is an open subset of $\Bbb{R}^n$) iff it is twice ...
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The averages of a subharmonic function over concentric balls increase with radius

Let $B_r$ a ball of radius $r$ in $\mathbb{R}^n$ and $u \in H^{1}(B_r)$ with $\Delta u =0$ in the weak sense. I am reading a paper and the author says that : "since $|\nabla u |^2$ is subharmonic, ...
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2answers
72 views

Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
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34 views

How to construct such a harmonic function on the upper half plane of $\mathbb{C}$ satisfying the following condition?

(1)Let u be a bounded harmonic function on the upper half plane of $\mathbb{C}$. Show that $\forall y$ we have $u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y\cdot u(t)}{(t-x)^2+y^2}dt$ for ...
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48 views

Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or ...
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2answers
38 views

Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define ...
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1answer
46 views

Holomorphic versus harmonic functions

Is it true that any holomorphic function on domain $D$ is of the form $u_x-iu_y$ for some harmonic function $u$? Motivation for this question is the problem of existence of harmonic conjugates.
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80 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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Why does $v(z)=\text{Im}\left[\left(\frac{1+z}{1-z}\right)^2\right]$ not contradict maximum principle?

Since $\left(\frac{1+z}{1-z}\right)^2$ is holomorphic in $\mathbb{D}$, its imaginary part is harmonic, and we have $$\underset{r \uparrow 1}{\lim}v(re^{i\theta})=0 \quad \forall ~ \theta \in [0, ...
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2answers
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What is an example of a nonconstant subharmonic function that attains a minimum?

Let $D$ be a domain in $\mathbb{C}$. What would be an example of a nonconstant subharmonic function that attains its minimum in the domain?
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2D Laplace Equation with Sine-squared BC

I am having a bit of trouble solving the 2D Laplace Equation $$\nabla^2u(y,z) = 0$$ with 2 BCs being $\left.\dfrac{\partial u}{\partial y}\right|_{y=0} = 0$ $u\left(y=\frac{h}{2},z\right) = ...
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Looking for pathologic counterexample: Nonzero harmonic function which is zero on the unit circle except 1

From my Complex Variables class: Let $C_1$ be the unit circle, $B_1$ the open unit disc and $\Gamma = C_1 \backslash \{1\} $. I am looking for a nonzero function $u \in C(B_1 \cup \Gamma)$ which is ...
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Prove $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic on the unit disc and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$ for all $\theta \in [0,2\pi)$. Explain why this does not ...
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1answer
81 views

Laplace equation in two dimensions and complex analysis

I was playing around with Laplace's equation: $$\frac{\partial^2 u}{\partial x ^2}+\frac{\partial^2 u}{\partial y ^2}=0$$ It occured to me that it can be written as: $$\bigg(\frac{\partial}{\partial ...
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97 views

Dirichlet problem on upper half disk

In our current Complex Variables homework assignment, we are given the following problem: Let $\Omega = \lbrace z \in \Bbb C : |z|<1, Im(z)>0\rbrace $ and $f \in C(\partial\Omega,\Bbb R)$ ...
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Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
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Prove ln(n) diverges as quickly as the harmonic series.

Question: Find a (simple) $f(n)$ so that $\lim\limits_{n \rightarrow \infty} \frac{n \sum\limits_{k=1}^{n} \frac{1}{k}}{f(n)} = 1$ My Attempt: I know the answer, by using Mathematica, is $f(n) = n ...
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Energy method for harmonic functions

I have two questions about the informations bellow that can be found in the book elliptic partial differential equations -QING HAN and FANGHUA LIN - chapter 1-pg 19 If $a_{ij} \in C(B_{1}(0))$ ...
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Comparison principle for functions $u,v$ such that $\Delta u-f(u)\ge \Delta v-f(v)$

Let $\Omega \in {{R}^{n}}$ and $u,v\in {{C}^{2}}\left( \Omega \right)\cap C\left( {\bar{\Omega }} \right),\text{ }f\in {{C}^{1}}\left( R \right)$ such that ${f}'\left( t \right)\ge 0$, for all $t\in ...
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How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
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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 ...
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55 views

The proof of Hopf lemma for harmonic functions

I would like to understand a passage from the proof of Hopf lemma. . In the second image above the author says: Therefore by theorem 1.29 (Maximum Principle for Subharmonic Functions) ...
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Harmonic non-negative function is constant

I'm having some trouble with the following: Let $u:\mathbb{R}^2\setminus\{0\}\rightarrow[0,\infty)$ be a harmonic function. Show that $u$ is constant. I have seen different proves for this. However, ...
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Steklov eigenvalue for the upper half plane

I have been thinking on this problem. Maybe someone know some previous work on this problem. The question is quite neat, consider the nonnegative solution $u\geq 0$ of the following equation ...
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Given $u$ subharmonic, show that $u^p$ subharmonic for $p\geq 1$

Exercise I'm currently reading through Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range, and I've been tasked with the following (Exercise II.4.4): ...
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Poissons formula from unit disk to upper half plane

I am trying to derive the Poisson's formula for the upper half plane from the formula on the unit disk using a conformal map. A conformal map from the unit disk to the upper half plane is (I think) $z ...
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62 views

Laplace $2$-D Heat Conduction

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space ...
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1answer
26 views

Proving harmonic function is zero

I'm having trouble with a homework assignment. This is the question: Suppose that $\Omega \subset \!R^3$ is a path connected bounded region and that $f : \Omega \rightarrow \!R$ satisfies $\Delta ...
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82 views

Sign of Laplacian Green's function in 3D

I am trying to prove that on a "nice" domain $\Omega$ in $\mathbb{R}^{3}$, the Green's function $G$ of $\bigtriangleup$ (the Laplacian) on $\Omega$ is always negative. I would like to use the Maximum ...
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Hardy-Littlewood theorem about the Poisson integral for $p=1$

(Hardy-Littlewood Theorem) : Let ‎$ u(r,‎\theta)‎$‎ be the Poisson integral of ‎$ ‎\varphi ‎\in L‎^{p}‎‎$‎, ‎$ 1<P ‎\leqslant‎ ‎\infty‎$‎ , and let $ U(‎\theta)=\sup‎_{r<1}‎|u(r,‎\theta)|‎ $‎. ...
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$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
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Alternate proof of Poisson Formula

I know how to prove this in general (I have a proof for Poisson formula I did), however, I have not been able to prove this way. Please help. Assume the theorem for $R=1$ deduce statement for any ...
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Neumann and Dirichlet Conditions for Schwarz-Christoffel Map

I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping. I'm able to map the polygon to the upper-half plane using ...
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1answer
33 views

Proving $v$ is harmonic

Let $u$ be a harmonic function in $\mathbb{R}^3$ and let $a > 0$. Show that the function $v$ defined in spherical coordinates by $v(r,\theta,\psi )=\frac{a}{r}u(\frac{a^2}{r},\theta,\psi)$ ...
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1answer
67 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 ...