Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
1,478
questions
0
votes
0answers
11 views
Laplace equation between circles with Dirichlet boundary condition
I am wondering if the below problem is solvable or not.
Laplace equation between two circle with raduses equal to $r=1$ and $r=r_0$ where $r_0<1$:
$\nabla^2 \phi(r,\theta)=0$
The boundary ...
1
vote
1answer
27 views
Can't retrieve Legendre equation from Laplace equation
I am tasked with the above, converting the 2-dimensional polar form of the Laplace transform:
$$
\frac{\partial}{\partial r}\Bigl(r^2 \frac{\partial z}{\partial r}\Bigr) +
\frac{1}{\sin(\phi)} \frac{...
1
vote
1answer
11 views
Determining function who satisfies Laplace equation.
Defining $\vec{r}=xi+yj+zk\ $ and $|\vec{r}|=r$, if $f$ is a scalar function such that $\nabla^2 f(r)=0$, then prove that $$f(r)=a+\dfrac{b}{r}$$where $a$ and $b$ are constants.
I found that $$\nabla^...
0
votes
1answer
21 views
Let $u$ be a harmonic function in $\mathbb{R}^n$ with $\int_\mathbb{R} |u|^p < \infty$. Then $u \equiv 0$.
This post Let $u$ harmonic. Then $\int_{\mathbb R^d}|u|^2<\infty \implies u=0$ answers this question in the case when $p=2$ using the Cauchy-Schwarz Inequality and an application of the Mean-Value ...
1
vote
1answer
29 views
question on Poisson equation with solution not in $H_0^1$
I'm considering the following quesion about Poisson equation:
$$
-\Delta u=f
$$
in a ball radius $1$ in $3$ dimension, if $f\in L^{2}$, then the
theory of elliptic PDE says that the above equation ...
0
votes
0answers
32 views
Math discrete: Proof by induction of harmonic serie
I need to prove by induction the following. The left hand side of the inequality is basically a harmonic serie where $H(1065^k)$. I've been able to complete this problem with $H(2^k)\ge 1 + \frac k2$, ...
0
votes
2answers
29 views
Closed form of series with the help of Harmonic series and generating functions
I am trying to find a closed form expression for the series:
$$ F(s) = \sum_{n>0} \frac{s^n}{n} \quad \text{for} \quad |s|<1$$
We know that we can define the harmonic number $ h_n = \sum_{i=0}^...
0
votes
0answers
23 views
Harmonic function on a ring
$\Delta f = 0$ in $X$ where $X=\{x \in \mathbb{R}^2:r_1 < |x| < r_2\}$.The boundary condition is $f(r_1,\phi)=v(\phi),f(r_2,\phi)=w(\phi)$.Solve a solution $f$.
My try:I used separation of ...
1
vote
0answers
20 views
when does bounded subharmonic function be a constant
I don't know why in 2-dimensional space, a subharmonic function bounded from above must be a constant while in higher dimensional space (when the dimension is larger than 2), the argument does not ...
0
votes
2answers
41 views
A priori error estimate for Dirichlet problem under geometric uncertainty
I am no specialist in PDE theory but I am interested where I could find an answer for the following question.
Consider two sets $D_1 \subset D_2 \in \mathbb{R}^d$ (actually only for $d \in \{2,3\}$) ...
0
votes
0answers
19 views
Kernel Matrix Calculation
Question:
Calculate the Kernel of:
$$F(x) = Ax$$
$$A=\begin{pmatrix}1 &-2& 4& 0\\
-3& -4& 10& -1\\
5 & 0& -2& 1\\
6 &-2 & 2& 1\end{...
0
votes
0answers
29 views
Lipschitz Domain and Integral Equation
I have the PDE problem
$$-\nabla^2u=0,\quad x\in\Omega$$
$$u=g,\quad x\in\partial\Omega$$
I have to figure out the integral equation involving the double layer potential for solving the above PDE. ...
0
votes
1answer
25 views
Laplace's equation in 2 dimensions with mixed Dirichlet and Neumann BCs: Is this the only solution?
I have a PDE that looks like this
$$\Delta T=\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}=0,\quad (x,y)\in \Omega$$
with boundary conditions
$$T(0,y)=20,\quad 0\leq y\...
0
votes
0answers
12 views
Extending harmonic function from half-space to whole space
$u$ is harmonic in $\mathbb{R}^n_+$ and $u=0$ on the boundary.
I wish to extend $u$ to a harmonic function on $\mathbb{R}^n$. Suppose I defined $u(x_1,...,x_{n-1},x_n)=u(x_1,...,x_{n-1},-x_n)$ for $...
0
votes
0answers
26 views
Comparison principle
This principle is from the harmonic function theory. Let $\Omega \subset \mathbb{R}^d$ be a bounded, open subset of $\mathbb{R}^d$ with compact closure $\overline{\Omega}$ for some $d \in \mathbb{N}$ ...
0
votes
0answers
24 views
Sum of squares of harmonic functions
I have to show $w(x)=\sum_ju_j^2(x)$ has its maximum value on the boundary $\partial\Omega$. The functions $u_j$'s are all harmonic. I think I need to show that $w$ is also harmonic and then I can use ...
0
votes
0answers
8 views
Weak Maximum Principle as a corollary of Strong Maximum Principle?
I am reading http://www.axler.net/HFT.pdf. The author proves the following :
$\Omega$ is an open connected set and $u$ is real-valued and harmonic in $\Omega$. If $u$ attains a maximum in $\Omega$ ...
0
votes
0answers
12 views
Showing that Green's function on a domain can be expressed in terms of Green's function on a conformally equivalent domain
First, this is the definition I have for Green's function on a simply connected domain: let $\Omega$ be a simply connected bounded open set and let $z_0 \in \Omega$. Then Green's function on $\Omega$ ...
0
votes
0answers
11 views
An easy way to solve this Poisson equation?
Problem : $-\Delta u=1-\sqrt{x^2-y^2}$ in $B_1(0)$;
$u=0$ on $\partial B_1(0)$
I could use the Poisson integral formula : $$u(x)=\int\limits_{B_1(0)}G(x,y)\Delta u(y)dy$$
where $G$ is the Green's ...
1
vote
0answers
30 views
Can complex analysis be used to solve Laplace's equation in three dimensions?
Can complex analysis be used to solve Laplace's equation in three dimensions? Or is it restricted to cases which have 2D symmetry?
All of the examples I've seen are 2D, never 3D.
It would seem that ...
3
votes
3answers
93 views
Liouville Theorem for Harmonic Functions
If $u$ is bounded and harmonic in $\mathbb{R}^n$, then $u$ is constant
For any twice differentiable function $u$ defined on an open subset $\Omega$ we have $$u(x)=\int\limits_\Omega G(x,y)\Delta u(y)...
3
votes
0answers
64 views
+50
Monotonicity formula for harmonic functions
Let $\Omega$ be an open subset of $\mathbb{R}^N$ and $u:\Omega\to \mathbb{R}$ be a harmonic function. I need to prove that, for every $x_0\in\Omega$, the function
$$
\rho\mapsto \frac{1}{\rho^N}\int_{|...
0
votes
0answers
14 views
Maximum Principle of Laplace equations
$u$ is the $C^2$ solution of
$$\begin{cases}\Delta u = 0 &\text{in }\mathbb{R}^d\backslash B_R\\
u = 0 & \text{on } \partial B_R\end{cases}$$
So the problem asks to show that ...
1
vote
1answer
34 views
How can we prove the harmonic mean is concave function?
For the function $f = \frac{n}{\sum_i 1/x_i}$ which is actually the harmonic mean of the values $x_1,x_2,\ldots,x_n$.
How can we prove this is concave function?
Because this is not a function of one ...
0
votes
1answer
13 views
$e^{jwt}$ is an ortho-normal basis proof
I'd like to know how one is supposed to show that the set $ \{ e^{jwt} \}$, where $\omega \in \mathbb{R}$ , is an ortho-normal basis?
So actually, how do I show that for every $w_1 \neq w_2 $:
$\...
2
votes
1answer
70 views
$\Delta u = 0$ in $B^{+}\cap \{x_{n} > 0\}$ and $u = u_{x_{n}} = 0$ in $\{x_{n} = 0\}$ $\implies $ $u = 0$.
Let $B^{+} = \{x \in \mathbb{R}^n ~ \colon ~ ||x|| \leq 1 ~~ \text{and} ~~ x_n >0 \}$ and $u \in C^2(B^{+})\cap C^1(\overline B^{+})$ such that
$$ \Delta u = 0 ~~ \text{ in } ~~ B^+ \hspace{3....
0
votes
0answers
35 views
bounded and uniformly continous harmonic function on the upper half plane
If $f(x)$ is bounded and uniformly continous on $\mathbb{R}$, and $u(x+iy)$ is harmonic function on the upper half plane $ {\mathbb{H}}$ defined by the convolution of the possion kernel and f(x), ...
3
votes
1answer
56 views
$u$ harmonic and $u^3$ harmonic then $u$ constant
Let $u(x,y)$ be a harmonic function defined in a connected open subset of $\Bbb R^2$. Does $u^3(x,y)=(u(x,y))^3$ harmonic implies $u$ is constant? It is easily shown as true, when I replace $3$ with ...
0
votes
0answers
5 views
Identity involving fundamental solution of Laplace equation
I have the following identity
If $\Omega$ is a smooth bounded domain and $u\in C^2(\bar\Omega)$ then for $x\in \Omega$ $$u(x)=\int\limits_\Omega \Phi(x-y)\Delta u(y)dy+\int\limits_{\partial\Omega}\...
1
vote
1answer
54 views
$|f|^{\alpha}$ harmonic for $\alpha > 0$
Wikipedia claims that given a holomorphic function $f$ then $\log(|f(z)|)$ being subharmonic implies that the function $\varphi_{\alpha}(z) := |f(z)|^{\alpha}$ is subharmonic for every $\alpha > 0$....
1
vote
1answer
21 views
How to estimate this integral?
In Evans's PDE, the section about Laplace's Equation, page 34 and page 36, I'm confusing about the following estimation: Given $U\subset\mathbb{R}^n$, $u(x)$ is harmonic in $U$, $v\in C^2(\bar{U})$, ...
0
votes
1answer
42 views
Smoothness proof for harmonic function
I was reading the proof of theorem 6 in Evans PDE. I do not understand last 2 steps in the proof. Please, can anyone help me to understand?
Any Help will be appreciated
0
votes
0answers
40 views
if $u$ is a harmonic function then $u$ has a harmonic conjugate.
I have a question about a theorem and its proof from the book, Functions of one complex variable(John B. Conway) 3.2 section.
Theorem:
Let $\pmb{G}$ be either the whole plane $\mathbb{C}$ or some ...
0
votes
0answers
51 views
Liouville's Theorem harmonic function
I'm working through Borthwick (2016) Intro to PDEs - specifically exercise 9.2. This exercise has 3 parts to it which all use Liouville's Theorem that all lead on from each other and I am having ...
0
votes
0answers
21 views
Difference between a Fourier series and a Harmonic series.
I have read that both these terms are used to mean a superposition of sinusoidal waves or functions with frequencies which are integral multiples of the frequency of the lowest frequency term (also ...
0
votes
0answers
25 views
Proof of the mean value property
Let $U\subset\mathbb{R}^n$ be an open and connected set and $u\in C^2(U)$ satisfying $-\Delta u=0, x\in U$. Moreover, let $x\in U$ and $r>0$ such that the open ball $B(x,r)\subset U$.
Let $h_x(v):=...
3
votes
2answers
85 views
Is there any way to calculate harmonic or geometric mean having probability density function?
I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like $K$, I have ...
2
votes
0answers
19 views
Behavior of Poisson Integral Formula for Half Space
I am trying to solve the following:
If $g$ is bounded and $g(x)=|x|$ for $x\in \mathbb{R}^{n-1}$ such that $|x|\leq 1$, then $Du$ is unbounded in a neighboorhood of zero.
Remembering that: the ...
1
vote
1answer
28 views
Proof for Mean Value Property using a specific limit
I am trying to prove the following:
Suppose $u \in C^2(\Omega)$. For some $x \in \Omega$ we have that
\begin{align} \Delta u(x) = \lim_{r \to 0} \frac{2n}{r^2} \left[
\frac{1}{w_n} \int_{\...
3
votes
1answer
39 views
About polynomial harmonic functions
I am really struggling to solve the following, I don't even know how to start. I would appreciate if anyone could give me some help.
Let $m$ be a positive integer and $u : \mathbb{R}^{n} \...
3
votes
2answers
68 views
Are harmonic functions always real analytic?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb{D}^n$. Let $f$ be a harmonic function w.r.t $g$.
Is it true that $f$ must be real-analytic?
I think that this ...
0
votes
0answers
31 views
Laplace problem for a circle with different type of boundary conditions
Let us consider the Laplace problem $\Delta u=0$ with the unitary circle as domain.
The boundary conditions are:
\begin{cases}
u=1 & \quad \text{if } \rho=1 \text{ and } \theta\in[\pi/...
1
vote
1answer
34 views
Finding A Function For The Harmonic series [duplicate]
Is it possible to formulate a function that can generate the next number in the harmonic series, for instance:
When $$ y = 4,$$ $$x=1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$
Thanks for your ...
0
votes
0answers
61 views
Summing The Following: $\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$
I can't seem to get this to sum, I'd be very thankful if someone could help me out.
$$\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$
NOTE It would be ...
2
votes
1answer
164 views
How to compute $\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$
How to compute
$$\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$$
where $\operatorname{Li}_2(x)=\sum_{n=1}^\infty \frac{x^n}{n^2}$ is the dilogarithm function.
...
1
vote
0answers
51 views
How to show a harmonic function is a polynomial?
Can anyone give me a hint to show the following?
Let $m$ be a positive integer and $u : \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a harmonic function. If $u(x) = O(\left|x \right|)^m$ when $\left|x \...
0
votes
1answer
50 views
Show that $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in the open disc with center 0 and radius 1
Show that the function $u(z)=\frac{1-|z|^2}{|1-z|^2}$ is harmonic in $K(0,1)$: the open disc with center 0 and radius 1, and the determine the conjugate harmonic functions to $u$.
I've been given the ...
0
votes
0answers
11 views
Local barrier can be extended to a barrier
If $\Omega, \Omega_1$ are bounded, open, connected subsets of $\mathbb{R}^n$ with $\Omega_1\subset \subset \Omega$ (alternatively, $\overline{\Omega_1}\subset \Omega$), $\zeta\in \partial \Omega_1$ ...
0
votes
1answer
29 views
$\Delta u = 1$ and $u$ convex $\implies$ $u$ quadratic polynomial
Let $ u \colon \mathbb{R}^{n} \to \mathbb{R} \in C^{4}(\mathbb{R}^{n}) $ be a convex function such that $\Delta u = 1$. Show that $u$ is a quadratic polynomial.
What I have done:
Every harmonic ...
0
votes
0answers
8 views
Is a solution of an elliptic PDE constant if all of its derivatives vanish at a point? (in analogy to unique continuation principle)
Consider a one-dimensional elliptic second order differential equation on an interval $[a, b]$.
I am specifically interested in Sturm-Liouville problems where the principal symbol is the Laplacian and ...
