For questions regarding harmonic functions.

learn more… | top users | synonyms (2)

0
votes
2answers
26 views

harmonic function. How to prove?

I've with prove if $1 \over |x|$ is a harmonic function. I know with for a harmonic function, $f_{xx}+f_{yy}=0$, but I don't know how to derivate ${1 \over |x|} dx$. And I don't know how to derivate ...
1
vote
0answers
29 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
4
votes
2answers
48 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 ...
5
votes
0answers
69 views

the real part of a holomorphic function on C \ {0, 1}

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

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
1
vote
1answer
38 views

Find a complex-differentiable function with real part $x^2(ay+8) +4y^2(y+b)$

Find a complex-differentiable function $f$ with real part $u(x,y) = x^2(ay+8) +4y^2(y+b)$ I have tried to use Cauchy-Riemann to get $v(x,y)$ but realised that I need to find the constants $a$ and ...
0
votes
1answer
31 views

Average Property of Harmonic Function

When we prove the average property of harmonic function, we use a formula \begin{align} & \int_{B_r(x)}\triangle u\,dy=\int_{B_r(x)}\text{div}(\triangledown u)\,dy \\[6pt] = {} & ...
0
votes
1answer
60 views

Zeros of a harmonic function

Prove that the zeroes of a Harmonic function is never isolated. All I can think of is a very rough idea of a proof by contradiction.
-1
votes
0answers
25 views

Harmonic function, inversion

Let $u$ be a harmonic function defined on $B_1(0)$, the unit sphere. Define $v: \mathbb{R}^d \setminus B_1(0) \to \mathbb{R}$ by $$v(x) = u(x/|x|^2).$$ How does one show that $v$ is harmonic?
0
votes
1answer
24 views

Gradient of Harmonic Function

Theorem If $u\in C(\overline{B_R(x_0)})$ and is harmonic in $B_R(x_0)$, then $$|D^mu(x_0)|\leq\frac{n^m\exp(m-1)m!}{R^m}\max_\limits{\overline{B_R~(x_0)}}|u|$$ We can prove the theorem by induction, ...
2
votes
2answers
38 views

Can Laplace's equation be solved in a domain that is not simply connected?

I have a problem where the domain is like a box with a tube missing - e.g. 0< x<1,0< y<1, 0< z<1 less the region (x-0.5)^2+(y-0.5)^2 < 0.25 In order to solve Laplace's equation ...
1
vote
1answer
40 views

Prove the Inverse of a Nonconstant Harmonic Function is Unbounded

Let $u$ be a nonconstant harmonic function on $\mathbb C$. Show that for any $c\in\mathbb R, u^{-1}(c)$ is unbounded. Hint: $\{|z|>R\}$ is connected for any $R>0$. It seems like this proof ...
0
votes
0answers
18 views

Subharmonic function equivalent non-negative laplacian

I want to ask for a proof that if $v(x,y)$ is $C^2$ and is subharmonic [here, define as satisfyingthen $\Delta v \geq 0$ where $\Delta v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 ...
1
vote
1answer
60 views

Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...
0
votes
0answers
20 views

How to check the barrier function is superharmonic?

Suppose $n\geq 3$ and $\Omega$ is a bounded domain. In the Perron's method to solve the PDE \begin{equation} -\Delta u = 0 \text{ in } \Omega \quad \text{and } u = g \text{ on }\partial\Omega, ...
0
votes
0answers
22 views

Fourier Transforms of hyperspherical harmonics

I am trying to compute the Fourier Transform of a function on a 3-sphere, $f(\hat{Q})$, where $\hat{Q}$ is a unit vector in four-dimensional space. The function $f(\hat{Q})$ is expressed as a series ...
1
vote
0answers
34 views

Show an equation only has harmonic solution

I want to show $$\begin{cases} \Delta(\Delta u) - \nabla\cdot (\Delta u \cdot \nabla u)=0\\ \int \Delta u < \infty\\ \Delta u \ge0 \end{cases}$$ in $\mathbb{R}^2$ only has a solution such that ...
0
votes
0answers
19 views

Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
11
votes
1answer
130 views

Source of the “$\cosh$ trick” for Laplacian eigenfunctions or Helmholtz equation solutions?

Suppose a smooth function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies the Helmholtz equation, the PDE $\Delta f + k^2 f = 0$. A while ago someone showed me a trick: Define a function ...
1
vote
0answers
37 views

Must a function hold true for all (x,y) to be harmonic?

I've found lots of examples that show various functions that are harmonic but I still can't figure something out. Does a function have to hold true for all (x,y) to be considered harmonic or is it ...
0
votes
0answers
14 views

By direct computation, check that the function u(x) = $|x|^{1/2}$ cos($\theta$/2) is harmonic in the upper half plane H := {x = (x1, x2) | x2 > 0}.

Okay so Partial Differential Equations make no sense to me. Don't know what to do. All I know is that if a function u is harmonic then $\Delta$u = 0
4
votes
1answer
73 views

Determining if a Continuous $u:\mathbb{C}\to \mathbb{R}$ Satisfying some Property is Harmonic

If $u : \mathbb{C} \to \mathbb{R}$ satisfies $$u(x + iy) =\frac{1}{4}[u(x + a + iy) + u(x − a + iy) + u(x + i(y + a)) + u(x + i(y − a))]\tag{$*$}$$ for all $a$ then determine whether $u$ is harmonic, ...
0
votes
1answer
26 views

Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
1
vote
0answers
22 views

Inverse Bessel Process as strict local martingale without Ito's formula?

Is there a way to prove that the inverse Bessel process $|B_t|^{-1}$ is a local martingale without using Ito's formula, considering that the Green's function $$g(x)=\int_0^\infty ...
0
votes
1answer
26 views

Describing Polynomials with Real Coefficients that are the Real Parts of Analytic Functions on $\mathbb{C}$

Describe those polynomials $$P=a + bx + cy + dx^2 + exy + fy^2$$ with real coefficients that are the real parts of analytic functions on $\mathbb{C}$. Idea (1): We are given that $P$ is the real part ...
1
vote
0answers
45 views

Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
0
votes
0answers
42 views

Reducing the Laplace equation with inhomogeneous BC's to the Poisson equation with homogeneous BC's

Given a domain $\Omega \subset \mathbb{R}^2$, one can reduce the Laplace equation $$\Delta u = 0, \qquad u = f \text{ on } \partial \Omega$$ to a Poisson equation $$\Delta v = g, \qquad v = 0 \text{ ...
1
vote
1answer
37 views

Finding the harmonic conjugate of $u(x,y)=\sinh(x) \sin(y)$

I know this is already a harmonic function but I am having trouble finding its harmonic conjugate. My instructor did this: $v_{x}=\cosh x \sin y \implies v(x,y)=\sin y \sinh(x)+g_{1}(x)$ ...
0
votes
1answer
26 views

2D Taylor expansion of F(x,y) where F(x,y) is harmonic (a solution of Laplace equation)

I would like to know if, given F(x,y) a real function of 2 variables that obeys $\nabla^{2} \left(F \left( x,y \right)\right) = 0$ , is it true that F(x,y) always equals its Taylor expansion within ...
6
votes
1answer
131 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 ...
3
votes
1answer
42 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
2
votes
1answer
30 views

Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
2
votes
2answers
71 views

Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
1
vote
0answers
43 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
0
votes
1answer
66 views

Why does from Perron-Frobenius follow that that constant functions are the only harmonic functions here?

in our reading we had the following example for a Markov chain. State Space $E=\left\{1,2,3,4\right\}$ and Transition Matrix $$ P=\frac{1}{3}\begin{pmatrix}0 & 1 & 1 & 1\\1 ...
1
vote
1answer
52 views

Extending a harmonic function satisfying a growth condition at an isolated singularity

Consider a harmonic function u on the punctured disc $\Delta(\rho)^*:= \{ z\in \mathbb C: 0 <|z|<\rho\}$ with $\lim\limits_{z \rightarrow 0} z*u(z) = 0$. Prove that $u$ can be written in the ...
0
votes
0answers
38 views

Equivalence for being subharmonic

I'm working with this definition of subharmonic function: Let $\Omega \subseteq \mathbb{R}^n$ be an open set. A function $u\in C(\Omega)$ is said to be subharmonic on $\Omega$ if for every ...
0
votes
1answer
52 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
0
votes
1answer
50 views

Green's function for Dirichlet problem on a half disk

Let $D=\{z=(x,y):x^2+y^2<R^2, y>0\}$ be the half disk with radius R. Then if we consider the Dirichlet problem on this domain, i.e., we want to find $$ \Delta u=0, ~~z\in D,\\ ...
1
vote
0answers
66 views

Poisson problem, green's function

I'm stuck at a Poisson integral problem and need some guidance. Assume that the Poisson integral is known as $$\int\int_S P(r,r')u_0(r')dr'$$ and gives the solution to the boundary value problems ...
1
vote
0answers
60 views

Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
1
vote
0answers
81 views

Solving Laplace equation in polar coordinates

I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them. Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , ...
0
votes
0answers
27 views

Proving $u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$ for harmonic $u$

I'm having a bit of a problem proving the equality: $$u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$$ Which is the mean value ...
0
votes
1answer
24 views

Holomorphic function locally represented as $(\partial_{x} - i \partial_{y}) h(x, y)$, with $h$ a scalar harmonic function

While reading an article, I came across the following statement. Moreover, locally every holomorphic function $f(x + iy)$ may be written as $(\partial_{x} - i \partial_{y})h(x, y)$, for some ...
2
votes
2answers
52 views

Limit of bounded harmonic functions is harmonic

I am trying to solve this old qual problem: Suppose $\{u_n\}_{n=1}^\infty$ is a sequence of functions harmonic on an open set $U \subset \mathbb{C}$ and uniformly bounded by 1. Suppose there is a ...
0
votes
1answer
88 views

Under which conditions is the harmonic function unique that has piecewise constant values on the boundary

$\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. $J_1 = \{e^{i\theta}: \theta \in (0, \pi/2)\}, J_2 = \{e^{i\theta}: \theta \in (\pi/2, \pi)\}, J_3 = \{e^{i\theta}: \theta \in (\pi, 2\pi)\}$ It's ...
0
votes
1answer
45 views

Convexity of sub-harmonic functions in a sector

Let $F(z)$ be an analytic function in an open sector $\Sigma_{\gamma}=\{0<\arg z<\gamma<\frac{\pi}{2}\}$, and continuous to the boundary. Then $\log{|F(re^{i\theta})|}$, $z=re^{i\theta}\in ...
0
votes
1answer
43 views

If $u$ is harmonic, prove that $|Du|^2$ is subharmonic.

We say $v \in C^2(\bar{U})$ is subharmonic if $-\Delta v \le 0$ in $U \subset \mathbb{R^n}$. Prove that $v := |Du|^2$ is subharmonic, whenever $u$ is harmonic. This is Exercise 5, part d, ...
0
votes
1answer
27 views

Mean-value formulas

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} ...
1
vote
1answer
32 views

Derivation of energy integral - harmonic functions

I am following the solution of the following problem on the topic of the energy integral of a surface. For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the ...