For questions regarding harmonic functions.

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23 views

Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
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1answer
38 views

Proove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function.

The problem is as follows: Proove that $U(x,y) = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ is the real part of an analytic function. where $f(z)$ is analytic such that ...
2
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1answer
54 views

Have I found my harmonic function correctly?

The problem statement is: Use conformal mapping to find a harmonic function $U(z)$ defined on the unit disc $∣z∣<1$ such that $$\lim_{r→1} U(re^{iθ}) = \left\{\begin{aligned} &+1 &&: ...
3
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2answers
39 views

Underdamped free vibration proof

I need to prove the solution form of: $$y''+2cwy'+wy=0$$ My book says, after assuming a solution of the form $Ce^pt$, you can show that: $$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$ I tried using the ...
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0answers
33 views

Subharmonic function.

Does anyone know where I can find properties involving integral average for a function that is subharmonic for heat equation, i.e., $$ u_t-\Delta u\leq0. $$ I need something like $$ u(x,t)\geq ...
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1answer
14 views

Helmholtz solutions on compact domains

Consider two compact domains $\Omega_1$ and $\Omega_2$ in $\mathbb{R}^n$, such that $\partial \Omega_1 \cap \partial\Omega_2$ is a real analytic hypersurface. Suppose I have an eigenfunction $\varphi$ ...
0
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1answer
34 views

Cylindrical Harmonics - Can't Find Bessel Equation

I'm solving Laplace's equation in a region with cylindrical symmetry (i.e. no polar angle dependence). Thus, from the outset, Laplace's equation becomes, $$ ...
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1answer
38 views

Maximum Modulus principle on Rudin's theorem.

I have studied rudin's Real and complex analysis, and have a question on the proof that why the level set $E$ is compact. Could you give me some hint? 11.13 Theorem If a continuous function $u$ ...
2
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1answer
33 views

Is subharmonic property preserved under mean value integral

In reading a paper, I've come across an "obvious" statment that looks funky. To make the question applicable to a wider audience let me simplify it a bit: Suppose $h$ is subharmonic on the ...
0
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0answers
22 views

External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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1answer
14 views

Smooth extensions from boundary of convex domains

Given a convex domain $\Omega$ with piecewise $C^1$ boundary data $g \in C(\partial \Omega)$ and $g \in C^1(\Gamma_i)$ with $\bigcup_{i} \Gamma_i = \partial \Omega$. Now, I want to know if there ...
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0answers
32 views

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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0answers
56 views

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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0answers
26 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 ...
3
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1answer
32 views

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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0answers
44 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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0answers
15 views

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
28 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{ ...
0
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1answer
44 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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0answers
21 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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1answer
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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1answer
23 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 ...
0
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1answer
56 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 ...
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0answers
25 views

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 $$ ...
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1answer
56 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 ...
0
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1answer
43 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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1answer
100 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
87 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 ...
0
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1answer
48 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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1answer
23 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 ...
0
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0answers
90 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 ...
2
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1answer
84 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 ...
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0answers
38 views

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 + ...
8
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1answer
198 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 ...
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1answer
49 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
35 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 ...
2
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1answer
35 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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0answers
61 views

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 ...
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1answer
70 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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1answer
28 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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1answer
38 views

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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1answer
54 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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3answers
122 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 ...
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1answer
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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2answers
134 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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1answer
65 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
150 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 ...
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0answers
25 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 ...
0
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1answer
39 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, ...