For questions regarding harmonic functions.

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PDE: Laplace equation Maximum Principle

The maximum priciple for Laplace equation assumes, in both PDE textbooks by Fritz John and Lawrence Evans, that the domain of the harmonic solution be bounded. Is the maximum priciple still valid if ...
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Fourier transform on Laplace equations

We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip. ; $y>0, -...
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28 views

Harmonic conjugates

Suppose $u$ is harmonic. We know by definition of chain rule that $$dv = \frac{\partial v}{\partial x} dx+\frac{\partial v}{\partial y} dy.$$ There is also a known formula of $$dv=-\frac{\partial u}{...
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Bounding harmonic functions.

Assume $f, g: \mathbb{R}^d \to \mathbb{R}$ are harmonic functions. Assume that there exist $C < \infty$ and $\alpha < 1$ such that for all $x$,$$|f(x)| \le C|x|^\alpha.$$What is the easiest ...
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62 views

Poisson Integral Formula for Upper Half Plane

It is known that the Poisson Integral Formula: $f(x,y) = \dfrac{y}{\pi}\displaystyle\int\limits_{\mathbb{R}}\dfrac{U(t)}{(x-t)^{2}+y^{2}}dt$ serves as a solution to the Laplace equation on the upper ...
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Weighted Analogue of Mean Value Property

Let $f$ be a harmonic function defined on a planar disk $B \subset \mathbb{R}^2$ and continuous on $\partial B$. For which probability measures $\mu$ on $\partial B$ is it true that $$\int_{\partial B ...
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1answer
33 views

Bounded solution to $\Delta u = f$ with certain boundary conditions tend to 0?

Let $T := \{(x, y) \in \mathbb{R}^{2}: x \geq 0, y \geq 0\}$ and suppose $f$ is a continuous function which vanishes when $x^{2} + y^{2} > R$ for some $R$. Suppose $u$ solves $$\Delta u = f \text{ ...
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1answer
50 views

Properties of a function of a harmonic function

Suppose that u : $\mathbb{R} \rightarrow \mathbb{R}$ is such that, whenever v : $\mathbb{R}^n \rightarrow \mathbb{R}$ is harmonic, so is u(v(x)). What can you say about u? My first inclination is ...
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31 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
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23 views

Show that the imaginary part of $\frac{z^2}{z-z_p}$ is harmonic

Let $z\in\Omega \subset\mathbb{C}$ and $z_p\notin \Omega$. Show that $\text{Im}(\frac{z^2}{z-z_p})$ is harmonic in $\Omega$, where $\text{Im}(z)$ is the imaginary part of $z$. So far: For $z = \alpha ...
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25 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
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91 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in \bar{U^+...
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asymptotic of an interesting recurrence realtion (more general case)

A link to the original question for reference:Click here I tried to study a more general situation: Let $y_{d,d}=1$ and $$ y_{n+1,d}=\frac{1}{n+1}\left(ny_{n,d}+\left(1+y_{n,d}-y_{n,d-1}\right)^{-n}...
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69 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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57 views

Given sequence of harmonic functions converges uniformly on compact subsets

Suppose that $u_n$ is a sequence of harmonic function on an open, connected subset $D \subset \mathbb{C}$ such that $u_n(z) \in (0, \infty)$ for all $z \in D$ and with $u_n(z_0) \to 0$ for some $z_0 \...
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126 views

Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
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3answers
111 views

Find a harmonic function in the first quadrant,

Find a harmonic function $\phi$(x,y) in the first quadrant with the boundary values $\phi$(x,0) = -1 for x>0, and $\phi$(0,y) = 1 for y>0. Is this function unique? My attempt was this: Consider ...
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1answer
53 views

$\Delta f=0$ in $\{x\in U:f(x)>0\}$ $\Rightarrow$ $\Delta f=0$ in $U$?

Let $f\geq0$ be a continuous function satisfying $\Delta f=0$ in $\{x\in U:f(x)>0\}$. I was wondering if one could follow $\Delta f=0$ in $U$, especially in the cases $f\in C^2$ or $\Delta f=0$ in ...
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25 views

Proof: superharmonic function equal on $\partial D$ and at one point inside of D to its harmonic function, is harmonic on D (D compact)

I am looking for a proof (literature or short idea) for the following statement, which I have found in several sources: Let $M$ be a riemannian manifold, let $f:M\to\mathbb{R}$ be a superharmonic ...
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119 views

Boundary conditions for laplace's equation for a rectangular box

Find the potential $\phi$, using Laplace's equation, inside a rectangular conductor that is subject to the boundary conditions $$ (i)\ \phi (x = a,y,z) = \cos(\beta y)\cos(\gamma z) \\ (ii)\ \phi (x,...
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96 views

Find a harmonic function in the interior of the disk, taking values +1 and -1

Consider the Mobius transformation $$f(z) = \frac{1+z}{1-z}$$ Use this map to find a function $f(x,y)$ which is harmonic in $x^2+y^2<1$ and on the boundary $x^2+y^2=1$ takes values $+1$ when $y>...
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Prove $u$ is non constant on each disk $D(r)$ for all $0 <r<1$ and $\varphi (r)$ is strictly increasing in $[0,1]$.

Let $D(r)=\{(x,y):x^2+y^2\le r^2\}$, and let $u(x,y)$ be a non constant harmonic function in the unit disk $D(1)$. Let $\varphi(r)=\max_{(x,y)\in\partial D(r)}u(x,y)$ for $0\le r\le 1$. Prove ...
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1answer
27 views

Deriving the Equation for the Depth to Which an Object Falls

For a body of mass m kg, show that the depth to which the body would fall if attached to a rope, with a length of l meters. The depth is given by the model: $$d= \frac{2ml \pm l \cdot \sqrt{4m^2 + ...
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237 views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator $\Delta^2=\Delta\left(\Delta\...
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1answer
51 views

A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...
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1answer
45 views

Bounded harmonic function on $\mathbb{R}^3$

Any suggestions how to get started? I know Liouville's theorem, but not sure how to apply it here: Let $u$ be a harmonic function on $\mathbb{R}^3$. Assume there exists $C>0$, independent of $...
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41 views

harmonic functions on the disk which agree on the real are identical?

The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please ...
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Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of real-...
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75 views

Describing the zero level set of a harmonic function

Let $p(k)$ be a complex polynomial of degree $n\in\mathbb{N}$. Let $A=\{k\in\mathbb{C}:\text{Re}\,p(k)=0\}$ The harmonic function $\text{Re}\,p(k)$ determines the behaviour of $A$. Fix $z\in A$. If ...
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30 views

Is the harmonic function constant?

Suppose $f$ is harmonic on $\mathbb{R^{2}}$ and constant on a neighbourhood in $\mathbb{R^{2}}$. Is $f$ constant on $\mathbb{R^{2}}$?
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Question on real-valued harmonic function

Let $V\subset\mathbb{C}$ be a connected open set and $u$ a real-valued harmonic function on $V$. Suppose that the set $S=\{p\in V \mid \nabla u(p)=0\}$ has a limit point in $V$, then $u$ is constant. ...
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55 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
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134 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question is:...
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72 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put $...
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230 views

Prove that a harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
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77 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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1answer
288 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary conditions:...
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35 views

Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
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46 views

Laplace equation in spherical coordinates

I am trying to calculate the Laplace equation($\Delta f =\partial_x\partial_xf + \partial_y \partial_y f + \partial_z \partial_z f = 0$ ) in $\Bbb{R}^3$ for spherical coordinates. $$g(r, \phi,\theta)...
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80 views

An application of strong maximum principle for harmonic functions

This should be an easy application, since it is given as a remark without proof in Evan's pde book. Let U be connected and u harmonic in U. Then u is positive everywhere in U if u is positive ...
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Solution procedure for poisson equation

Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$, $$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$ My Question: Is ...
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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:...
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251 views

Converse to mean value property: ball mean value property implies harmonicity

It is well-known that harmonic functions satisfy the mean value property. That is, if we set $\alpha(n)$ to be the volume of the unit $n$-ball, we have the following theorem. Let $u$ be an ...
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Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
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Show that a nonconstant subharmonic function on a manifold cannot attain its supremum

PROBLEM: Suppose $f$ is a smooth non-constant function on a connected Riemann manifold $M$ of dimension 2 such that $f$ is bounded and $\Delta_M f \ge0$. Show $f$ cannot attain its supremum. I try ...
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Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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37 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
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29 views

Problems identifying harmonic motion

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that ...
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67 views

Biharmonic operator; properties, identities

The biharmonic operator is $\nabla^4 \phi \equiv \nabla^2 (\nabla^2 \phi)$. Are there any identities for it? I need to find $\phi$ such that $~\\$ $\nabla^4 \phi = \frac{1}{3}\nabla^4 u^3 - u \nabla^...
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33 views

Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$ has a unique solution? Can ...