Tagged Questions

For questions regarding harmonic functions.

26 views

Trouble with an application of Green's representation formula

The teacher solved an exercise in class which required you to prove that, if $\Omega$ is a bounded domain in $\mathbb R^n$ and $G$ its Green function, then $G$ is symmetric, i.e. $G(x,y)=G(y,x)$ for ...
23 views

Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
23 views

48 views

$u$ and $u^2$ are harmonic.

Let $D$ be the unit disk centered at $0$ in the complex plane, and let $u$ be a real harmonic function on D. Find all $u$ such that $u(0)=0$ and $u^2$ is also harmonic on $D$.
16 views

Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
12 views

Removing Singularity of real bounded harmonic in punctured disk. [duplicate]

If $u(z)$ is real harmonic and bounded in the punctured disk $0<|z-z_0|<R.$ Show that $\lim_{z\to z_0} u(z)$ exists. I already know Complex analytic function $f$ which has singularity $z_0$ ...
14 views

Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
15 views

28 views

Does there exist a kernel concept for Taylor expansions?

In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ...
45 views

51 views

31 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.
94 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
71 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 ...
67 views

Proof that $Y=A\cos(px)+B\sin(px)$ is only periodic if $p=n$.

I am asked to show that a function $y=A\cos(px)+B\sin(px)$ can only be periodic if $p$ is an integer $n$, where $A$, $B$ are arbitrary constants. In other words $y(x)=y(x+2 \pi)$ I begin by solving ...
39 views

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 $... 2answers 163 views Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion? (First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ... 1answer 35 views Show$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 ... 2answers 67 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 1answer 25 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 ... 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 ... 1answer 46 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 $$\nabla ^2 G=\delta (\textbf x - \xi)... 2answers 60 views Why is \sqrt r \cos \frac \theta 2 harmonic? [closed] Why is \sqrt r \cos \frac \theta 2 harmonic? 1answer 74 views f is holomorphic in Ω such that |f|^2 is harmonic; we need to show that f is constant. f is holomorphic in Ω such that |f|^2 is harmonic; we need to show that f is constant. solution of the question In the solution attached, I don't really understand the transition between ∆|f(z)... 0answers 27 views Non-constant harmonic function satisfying given property Let u(x,y) be a non-constant harmonic function in region \mathbb{D}_{\mathbb{R}}=: D and$$A:=\{(x,y)\in D : u_x = u_y = 0\} $$what can one say about the set A? Since u is harmonic, there ... 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 non-... 2answers 36 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 u(x=a,... 1answer 19 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 ... 0answers 26 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 ... 0answers 22 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 C(\overline\... 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 \... 1answer 130 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 \mathrm{... 0answers 43 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 C_n=S_n\... 1answer 318 views Solving laplace's equation for an inviscid and incompresible fluid Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ... 1answer 85 views Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions Let \Omega be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem$$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), &... 0answers 61 views Tight bounds for harmonic measure I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ... 1answer 79 views a special extension of a two variable function We consider the function$f(x,y)=x^2+y^2$in$\omega = (0,1)^2.$I am wondering about the existence of a$C^2-$extension$F$of$f$in$\Omega = (0,2)^2$such that$F$is harmonic in$\Omega-\overline{...
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 \$\...
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 ...