For questions regarding harmonic functions.

learn more… | top users | synonyms (2)

0
votes
0answers
21 views

Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$ \Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
0
votes
0answers
25 views

How to prove a function is not positive definite

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
2
votes
0answers
24 views

$\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt$?

Prove that if $\zeta \in \mathbb{C}$ and $r>0$ then $\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt=\log |\zeta|$ if $r\leq |\zeta|$, and it is $\log r$ if $r> |\zeta|$. My Try: First I ...
-1
votes
0answers
32 views

stuck looking for a self similar solution to laplace equation [closed]

Please help, I've been stuck on this question for a while now: below is a solution to Laplace's equation $$\frac 1 \pi \left( \arctan \frac {x-a} y + \arctan \frac {x+a} y \right)$$ Limit as a self ...
0
votes
0answers
27 views

Prove: For a harmonic function $u$, $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ implies $u = 0$ for $0 < \epsilon < 0.5$

Prove: For a harmonic function $u: \mathbb{R}^3 \rightarrow \mathbb{R}$, if for all $(x,y,z) \in \mathbb{R}^3$ $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ then $u = 0$ for $0 < ...
1
vote
1answer
36 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on ...
1
vote
0answers
68 views

Is there a mistake in this proof in Rudin's RCA?

Rudin's Real and Complex Analysis, 3rd edition, page 236 : How did Rudin get to $$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$ It seems to me that he used the mean ...
14
votes
2answers
134 views

Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion?

(First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ...
0
votes
1answer
22 views

A radial harmonic function on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$

Prove: A radial harmonic function $f$ on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$ for $b,c \in \mathbb{R}$. My try: Label $g_i = (0,..,0,x_i,0,..,0)$. From the maximum ...
0
votes
1answer
23 views

Is it possible to analytically solve Laplace's equation between two rectangles?

I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below: Is it possible to do that analytically? So far I haven’t ...
-2
votes
2answers
59 views
0
votes
0answers
26 views

Non-constant harmonic function satisfying given property

Let $u(x,y)$ be a non-constant harmonic function in region $\mathbb{D}_{\mathbb{R}}=: D$ and $$A:=\{(x,y)\in D : u_x = u_y = 0\} $$ what can one say about the set $A$? Since $u$ is harmonic, there ...
0
votes
1answer
34 views

Show $u$ satisfy poisson equation

Let $f\in \mathcal S(\mathbb R^n),n\geq 3$. How can I show that $$u(x)=C_n\int_{\mathbb R^n}|x-y|^{2-n}f(y)dy$$ where $C_n$ is a suitable constant solve Poisson equation ? i.e. $$\Delta u=f.$$ The ...
3
votes
1answer
66 views

$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant.

$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant. solution of the question In the solution attached, I don't really understand the transition between ...
2
votes
1answer
12 views

Geometry of level sets of an harmonic function

Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open ...
0
votes
1answer
16 views

A question related to Laplace equation on pde.

Let $B_R(p)$ the open ball of radius R centered at p in $R^n$ and consider the following problem $∆u = 0 $ in $ R^n\setminus B¯_R(p)$ ,$u = c$ on $∂B_R(p)$, where c is a given constant. Find a ...
0
votes
2answers
33 views

What is a boundary condition for a PDE in a rectangular domain?

In the method of separation of variables, we need homogeneous BCs. For the elliptic pde with inhomogeneous BCs: $u_{xx}+u_{yy}=0$, with $0<x<a$ and $0<y<b$. With $u(x=0,y)=0$ and ...
0
votes
1answer
42 views

How to find the Green's function

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - ...
1
vote
0answers
25 views

Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
0
votes
0answers
21 views

Prove that a family of harmonic functions is a normal family

Suppose $\Omega$ is an open, bounded, connected set. Let $f$ be a continuous function on $\overline\Omega$ and $\mathcal{F}$ be the family of harmonic functions on $\Omega$ that belong to ...
0
votes
0answers
16 views

The harmonic functions are smooth

Let $u \in C^2(\Omega)$ a harmonic function, then $u \in C^{\infty}(\Omega)$. Here $\Omega$ is a domain of $\mathbb{R}^{n}$. Step 1 for proof: $u \equiv u^{\epsilon}$ in $\Omega_{\epsilon}$, for ...
1
vote
1answer
120 views

About a harmonic function in the upper half plane [duplicate]

I'm struggling with the following question: Suppose that $C$ is a positive constant, $u$ is harmonic in the upper half plane $\mathrm{Im}z>0$, and that $0 \le u(z) \le C\mathrm{Im}z$ for ...
5
votes
0answers
41 views

Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $S_n$ denote the discrete square $S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and ...
3
votes
1answer
36 views

$g: (U \times U - D) \to \mathbb{R}$ is continuous, $D$ diagonal? [closed]

Do we have necessarily have that$$g: (U \times U - D) \to \mathbb{R},$$is continuous, where $D$ is the diagonal? Idea. Perhaps we want to apply the maximum-minimum principle to $G(z, z_0)$?
1
vote
1answer
24 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
0
votes
2answers
21 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
0
votes
0answers
43 views

Equivalent definition of harmonic functions

Let $\Omega \subset \mathbb{R}^n$ be a domain. And let $u \in \mathcal{C}(\Omega)$. Prove that: The function u is harmonic in $\Omega$, i.e. $u \in \mathcal{C}^2(\Omega)$ and $\Delta u = 0$ on ...
1
vote
1answer
33 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r ...
0
votes
1answer
29 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
1
vote
0answers
36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
1
vote
1answer
20 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that ...
0
votes
0answers
13 views

Mean value property for harmonic functions (geometric idea)

Someone knows a bibliographic reference geometrically explore the idea of ​​the mean property for harmonic functions in domains of $\mathbb{R}^{n}$ ?
0
votes
2answers
36 views

Newton potential for Neumann problem on unit disk

Problem: Show that $$G(x, y) = \frac{1}{2\pi} \left( \log \lvert x - y \rvert + \log \left\lvert \frac{x}{\lvert x \rvert} - \lvert x \rvert y \right\rvert \right)$$ is a Green's function for the ...
0
votes
0answers
16 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
0
votes
2answers
108 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 ...
1
vote
1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
-1
votes
2answers
27 views

Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...
0
votes
0answers
17 views

Stuck while trying to show fundamental solution $\frac{1}{2\pi}\ln|x|$ satisfies $\Delta \phi(x) = \delta(x)$ in 2d?

I am trying to show that the fundamental solution to the Laplacian in 2D satisfies $$\Delta \phi(x) = \delta(x)$$ where $x = (x_1, x_2) \in \mathbb{R}^2$. So the fundamental solution in 2D is ...
2
votes
1answer
33 views

Properties of harmonic function on $\mathbb{R}^2$

Assume $f$ is harmonic on $\mathbb{R}^2$. I want to prove that if there exists a constant $M$ such that $f(x,y) \geq M$ for all $(x,y)\in \mathbb{R}^2$, then $f$ must be a constant fuction. I'm ...
3
votes
1answer
52 views

Computing $\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y$ using the mean value property.

I am asked to compute $$\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y,$$where $D = \{ (x,y) \mid (x+1)^2+y^2 \leq 9, \text{and }(x-1)^2+y^2 \geq 1 \}$. Granted, $u(x,y) = x^3-3xy^2$ is harmonic (it is the real ...
1
vote
0answers
28 views

What does it mean for a complex-valued function to be bounded above (or below)?

I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this: Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is ...
0
votes
1answer
24 views

Relation between Poisson kernel and harmonic measure

If $D$ is a domain in the complex plane bounded by a Jordan curve $J$, what's the relation between the harmonic measure and the Poisson kernel on the boundary? More specifically, if $z_0 \in D$ and ...
3
votes
1answer
97 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 ...
1
vote
1answer
39 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
2
votes
1answer
29 views

The value of a harmonic function in the interior of a unit disk

Let $u(z)$ be a bounded harmonic function in $D$ such that the limit $$\lim_{r→1^-}u(re^{iφ})$$ is equal to 1 when for $0 < φ < π$ and to 0 for $π < φ < 2π$. Find $u(1/2)$. ...
2
votes
1answer
59 views

Separation of variables for PDE: dividing by zero?

This feels like a question that is both simple and duplicate but I can't find an answer or a previous version of the question. Suppose we are given some PDE, for example Laplace's Equation in polar ...
0
votes
1answer
19 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
1
vote
0answers
18 views

3 dimensional harmonic conjugates?

An $n$ dimensional harmonic function is defined to be a real valued function $f$ in $\mathbb{R}^n$ such that $\nabla^2 f = 0 $. Equivalently, $f$ is the scalar potential of a conservative vector field ...
1
vote
1answer
79 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
0
votes
0answers
13 views

Critical points of a harmonic function

Suppose $\phi$ is harmonic on some compact, connected region of $\mathbb{R}^3$. Is there an algorithm that is guaranteed to find all critical points of $\phi$? (Obviously, these will all be saddle ...