For questions regarding harmonic functions.

learn more… | top users | synonyms (2)

0
votes
1answer
21 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 ...
2
votes
1answer
14 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
votes
1answer
23 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 ...
1
vote
0answers
18 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 $$ ...
4
votes
1answer
43 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
votes
1answer
20 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 ...
2
votes
0answers
13 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 ...
1
vote
3answers
41 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
votes
1answer
39 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 ...
1
vote
1answer
20 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
votes
0answers
23 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
votes
1answer
53 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 ...
1
vote
0answers
31 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 ...
0
votes
1answer
25 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 + ...
7
votes
1answer
158 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 ...
1
vote
1answer
48 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 ...
1
vote
1answer
28 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 $x$, ...
2
votes
1answer
29 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 ...
1
vote
0answers
33 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 ...
3
votes
1answer
57 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 ...
1
vote
1answer
25 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 $\mathbb{R^{2}}$?
0
votes
1answer
26 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. ...
1
vote
1answer
40 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. ...
1
vote
3answers
75 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 ...
0
votes
1answer
70 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 ...
1
vote
1answer
43 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
58 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 ...
0
votes
0answers
20 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
votes
1answer
31 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, ...
0
votes
1answer
41 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 ...
1
vote
0answers
35 views

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 ...
2
votes
0answers
23 views

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 ...
3
votes
1answer
31 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 ...
2
votes
1answer
16 views

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 $\Omega$?
4
votes
1answer
34 views

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 ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
19 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 ...
0
votes
2answers
22 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 ...
0
votes
0answers
45 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 ...
0
votes
1answer
25 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 ...
1
vote
1answer
23 views

Laplace equation in a circle with non-continuous Dirichlet boundary conditions

I have to solve: $$ \begin{cases} u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^2}u_{\theta \theta}=0 & [0,1) \times [-\pi,\pi] \\ u(1,\theta)=0 & (-\pi,0) & (1) \\ u(1,\theta)=1 & (0,\pi) ...
0
votes
1answer
20 views

non constant harmonic function

If $u$ is harmonic function on disk with radius $R$ around the origion, and non constant in it. why is it true that $u$ cannot be constant in any sub-Disk (i.e disk with radius less than $R$) thanks ...
0
votes
1answer
16 views

Methods for finding harmonic conjugate function

What are the methods for finding harmonic conjugate function? There is the cauchy - riemman equations but are there any other methods? Thank you very much
1
vote
0answers
23 views

Conformal transformation of a region bounded by a curve $y=x^a, a \in \mathbb{R}$

I would like to solve the 2D Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0$ on the positive upper half plane: $0 <x<\infty$ and $0 < y < ...
1
vote
1answer
53 views

How to solve 2D Laplace Equation over an infinite rectangular strip (bounded on two edges), with Dirichlet boundary conditions

Is it possible to solve Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$, over an infinite rectangular strip defined by $0 < x < \infty$ and $0 < y ...
1
vote
1answer
25 views

General solution of laplace equation

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? With many thanks for your help.
0
votes
0answers
7 views

about the mean value formula for harmonic functions

Let $ u \in C^{2}(B_R(x)) \cap C(\overline{B_R (x)}) $ a harmonic function. Does the function $u$ satisfies $$ u(x ) = \frac{1}{\omega_n R^{n-1}} \int_{\partial B_R (x)} u = \frac{1}{\omega_n ...
0
votes
0answers
31 views

Poisson's integral equation

Thank you. How can I find an harmonic function in the unit circle, that takes the value of \begin{equation*} F(\theta)= \left\lbrace \begin{array}{l} +T \text{ if } 0<\theta<\pi \\ ...
5
votes
1answer
170 views

Topology of solution to a nonlinear eigenvalue problem

Consider the elliptic PDE: $$-\Delta u= f(x) u. $$ Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial ...