0
votes
1answer
18 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
0
votes
0answers
32 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
1
vote
1answer
36 views

Regularity of weakly harmonic map

Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map ...
0
votes
1answer
39 views

Mean Value Property and Harmonic functions: a simple exercise

Let $f:[a,b]\to\mathbb{R}$ such that for every $h>0$ such that $$(x-h,x+h)\subset[a,b]$$ we have $$f(x)=\frac{1}{2}(f(x-h)+f(x+h)).$$ How can I conclude that $f$ is harmonic in $[a,b]$? My idea ...
0
votes
0answers
28 views

example of harmonic function on sphere

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren ...
0
votes
0answers
35 views

Chain Rule (applied twice) for vector valued functions

I need to show that if $f: \mathbb{R}^2 \to \mathbb{R}$ is harmonic, i.e., $$\frac{\partial^2 f}{\partial x^2}(x,y) + \frac{\partial^2 f}{\partial y^2}(x,y) = 0 \quad \text{ for all } (x,y) \in ...
0
votes
0answers
45 views

Second partial derivatives of harmonic functions

Given a twice-differentiable function $f:S\rightarrow \mathbb{R}$, where $S$ is a nonempty subset of $\mathbb{R}^n$, how does one prove that if f is harmonic then the second partials $D_jD_kf$ all ...
0
votes
1answer
41 views

Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
0
votes
1answer
34 views

Can we consider the points $(c,y)$ as local minima or maxima for all $y∈ℝ$

Let $f:(a,b)×ℝ→ℝ$ be a non zero and twice differentiable function. Let $c∈(a,b)$. Assume that $Δf=0$ (the Laplacian operator) and $f(c,y)=0$ for all $y∈ℝ$. Assume that $$f(x,y)<0=f(c,y)$$ for ...
1
vote
1answer
30 views

Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
1
vote
0answers
276 views

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 $x_0$, ...
2
votes
2answers
33 views

Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
0
votes
1answer
34 views

$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$. Any hint ?
1
vote
2answers
50 views

Harmonic functions constant on circumferences

I want to find all the harmonic functions in $\mathbb{R}^{2}-\{(0,0)\}$ which are constant on circumferences with center in $(0,0)$. $\mathbb{R}^{2}-\{(0,0)\}$ isn't simply connected so we can't ...
0
votes
1answer
52 views

Bounded (from below) harmonic functions from $\mathbb R^2 \setminus \{0\}$

Let $M \in \mathbb R$ be a real number and $u\colon \mathbb R^2 \setminus \{0\} \to [M, +\infty)$ be an harmonic function. Then it is constant. Show that this is no longer true in higher ...
1
vote
1answer
79 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 $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
1
vote
2answers
69 views

Show the Laplace Equation is rotationally invariant: Issues thinking about Laplace operator?

So I kind of get both methods of proof: http://math.gmu.edu/~memelian/teaching/Fall11/math678/hw/hw1sol.pdf But I'm having issues reconciling the definition of the Laplace operator as the sum of ...
0
votes
1answer
91 views

can the gradient of a harmonic function =0 at some interior point of a manifolds with two ends?

M is a complete noncompact Riemannian manifold with two ends. There exists a nonconstant bounded harmonic function f defined on the whole M. Then is it possible that $|\nabla f|=0$ at some interior ...
1
vote
1answer
120 views

About Mean Value Property of Harmonic Function

I know the question may seem foolish to you but I am not quite sure how to show it in a decent way. My problem is to show that for bounded Borel measurable $f:\mathbb{D}^2\to\mathbb{R}$, (D1) is ...
3
votes
1answer
119 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
4
votes
1answer
212 views

Proof of weak maximum principle.

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
8
votes
2answers
327 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 ...
1
vote
1answer
154 views

Harmonic functions

Let $f: \Omega \to \mathbb{R}$ be a harmonic function, where $\Omega \subset \mathbb{R}^2$ is an open subset. What can be said about the points where $\frac{\partial f}{\partial x} =\frac{\partial ...
1
vote
2answers
106 views

Is this function a subharmonic function?

Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$ for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
1
vote
0answers
128 views

Notation in Gilbarg/Trudinger? [Section 2.8]

Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
2
votes
0answers
287 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...
2
votes
1answer
146 views

Harmonic function with bounded preimage

I recently saw a question here about bounded/unbounded preimages of a set under a harmonic function. The question asked did not seem to make sense as it was talking about harmonic functions on ...
1
vote
1answer
83 views

Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded

Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded. I am not getting what theorem or result to apply. Could anyone help me?
0
votes
1answer
175 views

Asymptotic behavior of the fundamental solution of Laplace's equations

Consider the fundamental solution of Laplace's equation: $$\Phi(x,y):= \begin{cases} \frac{1}{2\pi}\ln\frac{1}{|x-y|},\quad x,y\in{\mathbb R}^2\\ \frac{1}{4\pi}\frac{1}{|x-y|},\quad x,y\in{\mathbb ...