For questions regarding harmonic functions.

learn more… | top users | synonyms (2)

-2
votes
0answers
7 views

x,y plane, divide up the x-axis by placing marks at x=c, x=b, and x = a [on hold]

In the x,y plane, divide up the x-axis by placing marks at x=c, x=b, and x = a. Suppose theta is harmonic in the upper half plane, and on the segments of the x-axis defined by your marks, theta takes ...
2
votes
2answers
22 views

On the partial derivatives of a harmonic function

Well, here is the thing. We know that the laplacian operator commutes with any partial derivative of a function, if the function is smooth. We also know that a harmonic function is infinitely ...
1
vote
1answer
26 views

Integrating a Poisson kernel in $n$ dimensional unit sphere

Let \begin{equation*} P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \end{equation*} be a Poisson kernel where $x$, $y$ are in $R^n$, $|x|<R$, $|y|=R$, $\omega_n$ is area of n dimensional ...
3
votes
1answer
38 views

Does a weaker form of the mean value property already imply harmonicity for continuous functions?

If $u:\mathbb{C}\to \mathbb{R}$ is continuous and satisfies $u(z)=\frac{1}{2\pi}\int_0 ^{2\pi}u(z+\frac{e^{i\theta}}{n})d\theta$ for all $n\in \mathbb{N}$ and $z\in \mathbb{C}$, is $u$ harmonic? What ...
1
vote
1answer
34 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
0
votes
1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
0
votes
1answer
36 views

laplacian of $1/\rho$ in cylindrical coordinates

In spherical coordinates, I believe that the laplacian of $1/r$ is zero everywhere except at $r = 0$ or \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi \delta^{(3)}({\vec{r}}). \end{align} where $r$ is ...
0
votes
1answer
14 views

Showing a function is harmonic on a domain - Imaginary part of $(A\cosh(z)+\frac\pi z)$

How to know that $\text{Im}(A\cosh(z)+\frac\pi z)$ is harmonic on domain $\{z|0\lt\text{Im }z\lt \pi\}$ where $A\in\Bbb R$? I am not sure how I would verify Laplace's equation here(which I imagine is ...
2
votes
1answer
30 views

Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where ...
1
vote
1answer
29 views

Proving that $\inf_{\nu}\varphi_{\nu}$ is subharmonic

Let $\Omega\subseteq\Bbb C$ open, $\varphi:\Omega\to[-\infty,+\infty[$ is subharmonic on $\Omega$ if $\varphi$ is uppersemicontinous on $\Omega$ For all $K\Subset\Omega$ compact, and for all ...
1
vote
2answers
17 views

An upper semicontinous function which is not subharmonic.

Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous. In ...
1
vote
1answer
21 views

Probability denisity function of simple harmonic

Suppose that a spring is oscillating up and down with vertical position given by $u(t) = \sin(t)$. If you pick a large number of random $t$ to look at the position, then prove that the PDF is ...
0
votes
1answer
15 views

Schwarz Reflection Principle for Harmonic Functions

Given $\Omega \subset \mathbb{R}^n$ define $\Omega^+ = \Omega \cap \{x_n>0\}$ and $\Omega^0$, $\Omega^-$ analogously let $u \in C^2(\bar{\Omega}^+)$ be harmonic and such that $\frac{\partial ...
0
votes
0answers
26 views

Harmonic Functions on Connected Open Set

From the maximum principle, any unbounded harmonic function $u : \Omega\rightarrow \mathbb R$ on a connected open set must be surjective. If $\Omega$ is bounded, does there always exist such a $u$?
0
votes
1answer
24 views

Harmonic function with vanishing partial derivative

Let $u:D(0,1)\to \mathbf{R}$ be harmonic on the unit disc, and suppose there exists a $z_0\in D(0,1)$ such that all partial derivatives of $u$ vanish. Show that $u$ is constant. I found this problem ...
0
votes
0answers
24 views

Prove one property of harmonic function

Let $u(x)$ be a harmonic function defined in the square $[0,1]\times[0,1]$. Suppose that $u(x_k)=0$, where $x_k=(1/k,1/k).$ Prove that $u(x)=0$ everywhere in $[0,1]\times[0,1]$.
0
votes
1answer
22 views

Surface Integral of the Partial Derivative of a Harmonic Function

Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$. Show that ...
0
votes
0answers
16 views

Quantum Harmonic Oscillators [migrated]

I'm having trouble with quantum harmonic oscillators and I'm not sure how to approach these questions: . I'd really like to get my head around these concepts but I'm struggling to understand fully. ...
2
votes
1answer
19 views

Link between harmonic and holomorphic functions on a non-simply connected domain.

There is a theorem that states that if a function $h$ is harmonic on a simply connected domain, there exists a holomorphic function $f$ such that $h = Re f$. Now, I am having a problem with the ...
1
vote
1answer
25 views

Intuition behind estimates on derivatives of a harmonic function

In Evans' PDE book he gives the following theorem. Assume $u$ is harmonic in $U$. Then, $$ |D^{\alpha}u(x_0) | \le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ When asking my professor for some ...
0
votes
1answer
19 views

The expansion of harmonic function at infinity

If $u$ is a harmonic function on $\mathbb R^n$ outside some compact set such that $u$ goes to $1$ at infinity. Then does $u$ have the following expansion $$ u=1+\frac{a}{|x|^{n-2}}+O(|x|^{1-n})\quad ? ...
6
votes
2answers
61 views

Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of ...
0
votes
0answers
20 views

What are the solutions of the Laplace equations for two (or more) eccentric cylinders?

I am looking for solutions to Laplace equation for two eccentric cylinders in 3D with arbitrary boundary conditions. The boundary condtions also depend on the axial variable. I tried to work with ...
2
votes
1answer
18 views

No Generalization of Mean Value Property for harmonic functions?

The Mean Value Property for harmonic functions tells us that the value of a harmonic function evaluated at the center of $D(P,r)$ equals its weighted integral over $\partial D(P,r)$. I am wondering if ...
0
votes
2answers
21 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
0
votes
1answer
41 views

If the integrals of a harmonic function over horizontal lines are uniformly bounded, it is identically zero

Let $u\colon\mathbb{R}^2\rightarrow\mathbb{R}$ be a harmonic function, such that $$\int\limits_{-\infty}^{+\infty} \lvert u(x,y)\rvert dx < C,$$ where $C>0$ is a constant not depending on ...
1
vote
1answer
20 views

Linear span of poisson kernels dense in $L^1(\mathbb{T})$

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions ...
0
votes
1answer
57 views

Proof of uniqueness for the Poisson equation

Show that the following problem has at most one solution: Given a continuous function $\rho(x,y,z)$ which is zero for $x^2+y^2+z^2>a^2>0$, find $\phi$ such that $$\nabla^2\phi=\rho$$ ...
2
votes
2answers
56 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
0
votes
1answer
30 views

Solution of the Laplace equation in polar coordinate.

Solve the following PDE: $$\phi(r,\theta) = \begin{cases} \Delta \phi=0 & \quad \text{for $a \le r\le b$ }\\[8pt] \phi=V & \quad \text{for $r=b$} \\[8pt] \phi+ C \sin(n\theta)=0 & ...
4
votes
2answers
35 views

Find a harmonic function in the cylindrical shell between $r=a$ and $r=b$

Calculate $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. I calculate it and found the solution is ...
2
votes
1answer
37 views

Solving the Laplace equation in a rectangle, using the separation of variables

Suppose I have $f_{xx}+f_{yy}=0$ on a region $R=\{(x,y):0\leq x\leq\alpha,0\leq y\leq\beta\}$ with boundary conditions $f(0,y)=f(\alpha,y)=0$, $f(x,0)=g(x)$, and $f(x,\beta)=h(x)$. I considered a ...
0
votes
2answers
31 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 ...
2
votes
0answers
39 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
59 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
1answer
85 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
26 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
39 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
35 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
80 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.
0
votes
1answer
26 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
49 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
46 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
36 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
62 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
32 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
23 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
35 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
20 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
132 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 ...