Articles of harmonic functions

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, then $u$ is constant in $U$. Proof: Let $x_0$ be an point in $U$ with $u(x_0)\geq u(x)$ for all $x$ in some neighbourhood […]

Reflection principle for harmonic functions

$U^+ \colon= \left\{x\in \mathbb{R}^n\mid |x| < 1, x_n>0\right\}$ is an open half-ball. Assume $u \in C^2 (\overline{U^+}$) is harmonic in $U^+$ with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. Set $v(x)=u(x)$ if $x_n \geq 0$ and $v(x)=u(x_1,x_2,\dots, -x_n)$ if $x_n < 0$ for $x \in U=B^0(0,1)$ . Prove $v \in C^2$ ($U$), and thus $v$ is […]

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive harmonic function $u(z)$ in $\Omega$ satisfies $u(z_2) \leq M u(z_1)$ for any two points $z_1, z_2 \in E$. This […]

Proving the weak maximum principle for subharmonic functions

Background Information: I am studying the book: Partial Differential Equations by Walter A. Strauss. I will be trying to adapt the proof given in chapter 6 section 1 to prove the following question. Definition: If $U$ is a bounded, open set, we say that $v\in C^2(U)\cap C(\overline{U})$ is subharmonic if $$-\Delta v \leq 0 \ […]

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 $$f(z) = u(x,y) + i v(x,y)$$ And I transformed what to prove into: $$u(\frac{\partial^2 u}{\partial^2 x} + \frac{\partial^2 u}{\partial^2 y}) + v(\frac{\partial^2 v}{\partial^2 x} + \frac{\partial^2 v}{\partial^2 y}) = 0$$ However, when I […]

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, it doesn’t make sense to solve Laplace’s equation with Dirichlet boundary conditions on $X$. That is, there does not exist any “valid”* $f$ such that $$\begin{align} […]

Harmonic function in a unit disk with jump boundary data

I am reading Conway’s book about complex analysis. One question in it bothered me a lot recently. If given a piecewise continuous function with jump on the boundary of unit disk and it is bounded, we can define a function in the disk by using Poisson kernel. Is this function still harmonic as in the […]

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. First, I assume $ f $ is an analytic function s.t. $ u(x, y) = \text{Re}(f(x, y)) $. Then, I used this […]

Show that the Poisson kernel is harmonic as a function in x over $B_1(0)\setminus\left\{0\right\}$

Show that the Poisson-kernel $$ P(x,\xi):=\frac{1-\lVert x\rVert^2}{\lVert x-\xi\rVert^n}\text{ for }x\in B_1(0)\subset\mathbb{R}^n, \xi\in S_1(0) $$ is harmonic as a function in $x$ on $B_1(0)\setminus\left\{0\right\}$. On my recent worksheet, this task is rated with very much points. So I guess it is either very difficult or requires much calculation. Am I right that I do have to […]

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 $h:\overline B\to\mathbb{R}$ which is continuous on $\overline B$, harmonic in $B$, and satisfies $u|_{\partial B}\leq h|_{\partial B}$, then $u\leq h$ on the whole $\overline B$. No regularity assumed for $u$, except for […]