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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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6
votes
4answers
1k views

A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$

My problem is to integrate this expression: $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$. I know the answer is zero. Can you explain you idea to me or just prove ...
14
votes
4answers
7k views

Composition of a harmonic function with a holomorphic function is still harmonic

If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ is a conformal mapping of a domain $D_0$ onto $D$, is $f \circ g$ harmonic in $D_0$? I noticed this question while reading ...
10
votes
1answer
3k views

If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity

This is a reworking of a previous question here which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the ...
5
votes
1answer
1k views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
4
votes
1answer
729 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 $C>0$...
2
votes
3answers
3k views

Positive harmonic function on $\mathbb{R}^n$ is a constant?

Is it true that a positive harmonic function on $\mathbb{R}^n$ must be a constant? How might we show this? The mean value property seems not to be the way...for that we would need boundedness.
1
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1answer
2k views

Maximum principle for subharmonic functions

Let $\Omega$ be a domain of $\mathbb{R}^n$, and $u:\Omega\to\mathbb{R}$ a continuous function. We call $u$ subharmonic if for any ball $B\subset\subset\Omega$ and any $h:\overline B\to\mathbb{R}$ ...
2
votes
2answers
2k views

How to prove Liouville's theorem for subharmonic functions

I noticed this post and this paper, which gives a version of Liouville's theorem for subharmonic functions and the reference of its proof, but I think there must be an easier proof for the following ...
8
votes
2answers
547 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set $U$...
7
votes
1answer
2k views

Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
2
votes
1answer
2k views

Show that the Kelvin-transform is harmonic

First, I have to give you our definitions: Consider $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. The function $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
3
votes
1answer
380 views

Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant. [duplicate]

Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant. I think i should use Liouville's theorem, but how can i ...
1
vote
1answer
184 views

Proving a function is harmonic without calculating partial derivatives

This is for an assignment, so I don't want any explicit solutions, but just some kind of idea how to continue. I'm trying to prove that the family of functions $u_{\xi}(z)=\frac{1-|z|^{2}}{|\xi-z|^{2}...
0
votes
1answer
978 views

Invariance of subharmonicity under a conformal map

I want to prove the following (exercise from Ahlfors' text): Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping. Here is my ...
7
votes
3answers
443 views

Bounded, non-constant harmonic functions: how far are they from existing?

Let $f$ be a function that maps $\mathbb{Z}^2$ to $\mathbb{R}$ and consider the operator $T$ which replaces the value of $f$ at $(i,j)$ by the average of the values of $f$ at its four neighbors: $$ Tf(...
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 ...
12
votes
1answer
3k views

Weakly Harmonic Functions (Weak Solutions to Laplace's Equation $\Delta u=0$) and Logic of Test Function Techniques.

In analysis we often use test functions $\phi\in C_{0}^{\infty}(U)$ in order to make some kind of deduction about another function $u:U\mapsto\mathbb{C}$. For example, if one can obtain the ...
10
votes
2answers
993 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of $u$...
7
votes
2answers
1k views

Prove that a non-constant 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 ...
10
votes
1answer
906 views

Harmonic functions with zeros on two lines

For which pairs of lines $L_1$, $L_2$ do there exist real functions, harmonic in the whole plane, that are $0$ at all points of $L_1 \cup L_2$ without vanishing identically? Note: This is self-study -...
5
votes
1answer
2k views

Laplace equation Dirichlet problem on punctured unit ball.

Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering ...
2
votes
2answers
2k views

Derive the Poisson Formula for a bounded C-harmonic function in the upper half-plane.

My book gives the Poisson Formula for such a harmonic function as: $$ u(x + iy) = \frac{1}{\pi} \int_{-\infty}^{\infty}{\frac{y \cdot u(t) dt}{(t - x)^2 + y^2}} $$ Here is what I have attempted. ...
5
votes
1answer
219 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
5
votes
1answer
2k views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial n}=\...
5
votes
1answer
793 views

Show $\Omega$ is simply connected if every harmonic function has a conjugate

Prove: If every harmonic function on $\Omega$ has a harmonic conjugate on $\Omega$, then $\Omega$ is simply connected. The same question is asked here but no proof is presented: is the converse true: ...
5
votes
1answer
1k views

You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
1
vote
3answers
568 views

Finding Harmonic conjugate for $\arg(z)$

Let $u(z)=\arg(z)$ in $D=\mathbb{C} \setminus\mathbb{R}^-$ where $\mathbb{R}^-=(-\infty,0]$. Find Harmonic conjugate for $u$. Any ideas / hints?
11
votes
2answers
5k views

Logarithm of absolute value of a holomorphic function harmonic?

Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$. Is it true that $z\mapsto \log(|f(z)|)$ is ...
6
votes
1answer
739 views

Interior gradient bound

I would like some help with the following problem (Gilbarg/Trudinger, Ex. 2.13): Let $u$ be harmonic in $\Omega \subset \mathbb R^n$. Use the argument leading to (2.31) to prove the interior ...
4
votes
1answer
416 views

A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
4
votes
2answers
126 views

Prove $\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|u|^2$ if $-\Delta u(x)+f(x)u(x)=0.$

Let $\Omega $ a smooth domain of $\mathbb R^d$ ($d\geq 2$), $f\in \mathcal C(\overline{\Omega })$. Let $u\in \mathcal C^2(\overline{\Omega })$ solution of $$-\Delta u(x)+f(x)u(x)=0\ \ in\ \ \Omega .$$ ...
3
votes
1answer
209 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
3
votes
1answer
385 views

Two question on harmonic function

In a question paper I got the following two questions. $u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex ...
1
vote
2answers
206 views

Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define $...
1
vote
1answer
961 views

Fourier Transform - Laplace Equation on infinite strip - weird solution involving series

I need to solve the following problem on the infinite strip: $\displaystyle \begin{align} u_{xx}(x,y) + u_{yy}(x,y) = 0, & -\infty < x < \infty, & 0<y<1 \\ u(x,0)= \begin{cases}1, ...
3
votes
1answer
100 views

Prove that $(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$

Knowing: $f(z)$ is analytical Prove: $$(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$$ I have proved firstly that $\ln|f(z)|$ is harmonic function Let $$f(...
1
vote
1answer
453 views

Using Green's identity to show that a harmonic function with zero boundary values is identically zero

I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected. I have subbed in the information ...
1
vote
0answers
243 views

Determining the Asymptotic Order of Growth of the Generalized Harmonic Function?

How should I proceed to determine the order of growth of the generalized harmonic numbers? $$ H_{n}^{(r)} \in \mathcal{O}(?) $$
0
votes
1answer
167 views

Choose parameters to make a harmonic function

Let $B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ |x|=\...
12
votes
1answer
1k views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
18
votes
2answers
2k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
10
votes
2answers
2k views

Show harmonic function is constant on $\mathbb{R}^n$

I'm trying to solve the following question (this is just for practice): If $u$ is harmonic within $\mathbb{R}^n$ with $\int_{\mathbb{R}^n}|Du|^2 dx \leq C$ for some $C > 0$, then show that u is ...
5
votes
2answers
1k views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\...
3
votes
1answer
2k 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 v}{\...
3
votes
1answer
708 views

Proving the Mean-value theorem in Evans.

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{...
9
votes
2answers
7k views

How do you prove that $\ln|f(z)|$ is harmonic?

Suppose that $f(z)$ is analytic and nonzero in a domain $D$. Prove that $\ln|f(z)|$ is harmonic in $D$. I know the laplacian equation but I'm not sure how to use it.
5
votes
1answer
514 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 ...
4
votes
1answer
257 views

is there a way to prove the Mean-value formulas using complex analysis?

THEOREM : $U$ is an open subset of $\mathbb{R^n}$ and suppose $u \in C^2(U) $ is harmonic within $U$, then : $$u(x)= \def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{B(x,r)}u\,dy = \def\...
2
votes
3answers
728 views

$f$ is analytic, nonzero on simply connected domain. Show that $\log|f(z)|$ is harmonic.

$f$ is analytic, nonzero on a simply connected domain $\Omega \subset \Bbb C$. Show that $\log|f(z)|$ is harmonic on $\Omega$. I thought of two methods: $f=u+iv$ so $\log|f(z)|=\log\sqrt{u^2+v^2}$ ,...
5
votes
3answers
155 views

$f \in C^2(\mathbb{R}^2)$ is harmonic; $f$ is in one $L^p(\mathbb{R}^2)$ $\implies$ $f$ is constant

Suppose $f \in C^2(\mathbb{R}^2)$ is harmonic. Suppose that $f$ is in one $L^p(\mathbb{R}^2)$ space, with $1 \le p \le \infty$. How can I prove that $f$ is constant and in particular is identically $...