The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

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Kato inequality

For any real-valued smooth function $u$, we have the Kato inequality $|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$, which holds when $|\nabla u|\neq0$. If moreover $u$ ...
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The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
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74 views

is Poisson's kernel always integrable?

Let $E$ be a smooth domain. The Green function $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation and for fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function in ...
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71 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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439 views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
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71 views

Maps in $\mathbb{R}^n$ preserving harmonic functions

A map $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ preserves harmonic functions if $f\circ\varphi$ is harmonic for every harmonic function $f:\mathbb{R}^n\to\mathbb{R}$. It is known that these maps are, in ...
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774 views

Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the ...
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235 views

Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
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Partial Differential Equations Exam help

new poster here. I have an exam in a PDE course in 72 hours which uses the second edition of Strauss. I am pretty scared because I don' exactly understand what I am doing. For instance, Chapter 6 ...
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94 views

Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
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24 views

For which real constants $k$ is the function $u$ harmonic?

For which real constants is the function $$ v(x):=x_1^3+kx_1x_2^2 $$ harmonic on $\mathbb{R}^n$? To my calculation, the equation $$ \Delta v=\sum_{k=1}^n\frac{\partial^2 ...
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25 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
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115 views

Gradient of an harmonic function

Let $ M $ be a Riemannian manifold and let $ f $ be an harmonic function on $ M $. By Unique continuation theorem we can assert that if $ \nabla f = 0 $ on an open subset $ \Omega \subset M $ then ...
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70 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
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72 views

Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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42 views

For compact $K \subset \mathbb C$, show that $u(z) = -\log(\mathrm{dist}(z,K))$ is subharmonic

Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic? Subharmonic, here, is ...
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139 views

Biharmonic operator

Consider the problem: $$ \Delta^2 u = f$$ on the square domain $U=(0,1)\times(0,1)$ with boundary conditions: $$ u(x,y)=\Delta u(x,y) = 0$$ for $(x,y) \in \partial U.$ I try to solve it with the ...
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A function in the $L^2$ closure of the set of smooth, harmonic functions on the closed unit disk is smooth and has a harmonic representative.

This is is a homework problem I'm having trouble understanding. I am given the set $$Y=\{u\in \mathbb{C}^{\infty}(\bar{D_1})| \triangle u=0\}$$ and its closure with respect to the $L^2$ norm, ...
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49 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 ...
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39 views

hyperbolic group ; showing the existence of a ration function with a certain condition

I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost. Right now I'm looking at an exercise (12.5) under the ...
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232 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 ...
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51 views

Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
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Derivatives of the Green's Function

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. ...
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Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
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31 views

Harmonic Function With Step Function Boundary Data

Consider the Unit Disk. can we solve for a harmonic function in the unit disk such that: $\triangle u = 0 $ in D and $ u = f $ on $\partial D$ where $ f = 1$ for $|\theta| \leq \epsilon$ and $ |\theta ...
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74 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 $x_0$, ...
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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Biharmonic boundary condition

I try to solve $$\Delta^2u=f$$ on unit square. with $f=4sin(\pi x)sin(\pi y)$ Using $v=-\Delta u,$ leads to $$v+\Delta u=0,$$ $$-\Delta v=f.$$ By Dirichlet boundary condition on $u$. What is ...
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What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
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298 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
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Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
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53 views

a version of the comparison lemma (harmonic functions)

I am trying to solve this exercise : Consider $U$ open , bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U)$ a harmonic function. Suppose that for each $x \in \partial U $ there exists a ...
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38 views

How to prove a harmonic function to be smooth?

With the Green formula, we can divide a function into several potentials. Being harmonic, there should be two, double and single layer. With this, how to prove a harmonic to be smooth?
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Geometric Condition for Harmonic Function

Under what geometric condition on real harmonic functions u and v on a region G is the function uv also harmonic?
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419 views

Green's function for the Laplace operator $\Delta$ in a rectangle (or square)?

What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm ...
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Harmonic Function in $\Omega$ that is continuous in $\overline{\Omega}$ except at a point on the boundary

My problem is the following. Let $x_{0}\in\partial\Omega$ and $\Omega\subseteq\mathbb{R}^{2}$ open and connected domain. Suppose there exists $R\in\mathbb{R}$ such that $\Omega\subseteq B_{x_{0},R}$. ...
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Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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$|Du|^2$ is subharmonic if $u$ is harmonic.

In Evan's textbook "Partial Differential Equation", question 5 in section 2.5 says "$|Du|^2$ is subharmonic if $u$ is harmonic.". This can be easily proven, but do we really need the derivative $D$? I ...
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127 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 ...
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201 views

Poisson equation on half-space

Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$. Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$. Define the function $F : \Bbb R^n \to \Bbb R$ by $$ ...
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Suppose f(z) = u + i v is non-constant and analytic on a domain D

Suppose $f(z) = u + i v$ is non-constant and analytic on a domain $D$. Which of the following statements, if any, is true? (1) $3u^2 + v^2 +uv$ is harmonic on $D$; (2) $u^3 + 3u^2 - 3v^2 - 3uv^2 + ...
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Looking at Green's Function for the Dirichlet Problem, how do you calculate n(y)

If G(x,y) = (-1/4pi)[(|x-y|^-1)-(|x-r(y)|^-1)] where r(y)=(y1,y2,-y3) and x and y are vectors in R3. The question states 'Evaluate ∂G(x,y)/∂ny (n subscript y) for y ∈ ∂Ω. I know to do this you ...
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Analytic property of harmonic functions through mean value property.

I have this strong intuition that one should be able to at least prove that a harmonic function is infinitely differentiable, let alone analytic, using just the mean value property. Unfortunately I am ...
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23 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 ...
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37 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 ...
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Bounding high-order partial derivatives of a bounded harmonic function

Fix $d\ge 2$ (integer). For any $x\in\mathbb{R}^d$ and $r\ge 0$, let $B(x,r)$ be the open ball of radius $r$ around $x$ in $\mathbb{R}^d$. I wish to show that given a constant $M\ge 0$, if ...
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36 views

Stationary heat equation problem

We are given the following BVP: $u_{xx}+u_{yy}=0,\ x\in\mathbb{R},\ y>0\\ u(x,0)=f(x),\ x\in\mathbb{R}\\ u(x,y)\rightarrow 0\ \text{as}\ x^2+y^2\rightarrow\infty$ where ...
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47 views

where $\nabla^2V = 0$ , evaluate $\int_S V d\Omega /4\pi$

Where $\nabla^2 V = 0$ in 3 dimensional Euclidean space, it is a well-known fact that $${\int_S V(\vec{r'}) d\Omega'\over 4\pi}=V(\vec{a})$$ where $\vec{a}$ is the center of a sphere $S$ of radius ...
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61 views

Harmonic motion, contract cosinus/sinus expression

I got a harmonic motion given as three partial motions, I am to write the function on the form of: $H(x)= A \cos(\omega(x-x_0)$ The function given is: $H(x) = ...