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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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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Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...
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Dirichlet Problem on the unit disk

Find a C-harmonic function in the unit disk with boundary values $x^3-xy$. I know the answer is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$ but don't know how to solve it Any hint or help is ...
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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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144 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
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112 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
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1answer
98 views

Upper bound for coefficients of a power series

I am doing the following problem. Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function on the unit disc $|z|<1$. Let $0<r<1$. Prove that $$|a_n|r^n\leq \max\{4A(r),0\}-2Ref(0),$$ ...
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167 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
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1answer
77 views

Harmonic function reflection

I'm learning some harmonic function theory by reviewing some problems. I came across two: 1) Prove that a real harmonic function $u$ from $\mathbb{R}^n$ to $\mathbb{R}$ such that $u(x, 0) = 0$ for ...
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41 views

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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1answer
171 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
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1answer
321 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
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462 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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1answer
142 views

Laplace equation on unbounded set

I have gotten stuck with a problem for PDEs class for a few days. I did not figure out how to start a solution for it. Problem: Let $g \in C(\partial B(0, R))$, $n > 2$. Find a formula for a ...
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1answer
64 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
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1answer
204 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 $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
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2answers
84 views

The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
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1answer
336 views

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 $\partial U^+ \cap \{x_n=0\}$. ...
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1answer
149 views

poisson's equation with robin's boundary, boundary value problem

Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...
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81 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
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1answer
125 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
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1answer
59 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
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Problem of Harmonic function.

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an ...
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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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178 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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2answers
582 views

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 ...
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Is this Harmonic Polynomial Identically Zero?

Let $h$ be a harmonic polynomial that is zero on the lines $Im(z) = 1$ and $Im(z) = -1$. I know that by the Schwarz Reflection Principle, $h$ must be zero on any line $Im(z) = k$ for $k$ odd, but ...
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1answer
176 views

What is relationship between Wirtinger differential operator and multivariable chain rule?

What is relationship between Wirtinger differential operator(equation 5) and multivarible chain rule(equation 4)? for other Wirtinger related questions look here.
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344 views

Positive harmonic function on $\mathbb{R}^n$ is a constant?

Is it true that a positive harmonic function on $\mathbb{R}^n$ must be a constant? How might we show this? The mean value property seems not to be the way...for that we would need boundedness.
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How to prove $u(x,y)=x^2-y^2+2y$ is harmonic? [on hold]

What is a harmonic function? How do you prove that a function is a harmonic?
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1answer
58 views

square of a harmonic function bound

I need to solve this problem: Let $u$ be a harmonic function inside the open disk $K$ centered at the origin with radius $a$. We are also given that $\int_K u^2(x,y)dxdy=M<\infty.$ Show that ...
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1answer
51 views

Boundary data of the modulus of a holomorphic function

Let $f$ be a non-vanishing holomorphic on the unit disk $D$. Suppose $|f|$ converges to a measure $\mu$ on $\partial D$ as $|z|\rightarrow 1$, in the sense that $$ \int_{\partial D} |f(r z)| \phi(z) ...
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92 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 ...
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1answer
160 views

Harmonic function takes both positive and negative values

I am a little confused on the following question: Suppose that $u$ is harmonic nonconstant on a $D(z_0,R)$ and $u(z_0)=0$. Is it true that on each circle $C(z_0,r)$, with $0<r<R$, the function ...
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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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Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized

My professor mentioned something like "Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized." I've been trying to understand this statement. If I say that ...
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1answer
108 views

Subharmonic functions in the punctured disk

I want to prove the following (exercise from Ahlfors' text): If $\Omega$ is the punctured disk $0<|z|<1$ and if $f$ is given by $f(\zeta)=0$ for $|\zeta|=1$, $f(0)=1$, show that all ...
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246 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
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55 views

Harmonic function on upper-half space

Let $H=\{(x,y,z)\in\mathbb R^3\,|\,z\geq 0\}$, let $f:H\to\mathbb R$ be harmonic on the interior of $H$, and let $f$ satisfy the boundary condition $f(x,y,0) = a$ for some $a\in\mathbb R$. One easily ...
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1answer
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Is my proof correct? (Invariance of subharmonicity under a conformal map)

I want to prove the following (exercise from Ahlfors' text): Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping. Here is my ...
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1answer
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Is my solution correct? (subharmonicity of several functions)

I want to show that the functions $|x|,|z|^\alpha(\alpha \geq0),\log(1+|z|^2):\mathbb C \to \mathbb R$ are all subharmonic. (This is an exercise from Ahlfors' text) $|x|$ $x$ and $-x$ are ...
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1answer
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Is there a harmonic function which satisfies the following conditions?

Let $\Omega\subset R^2$ be a simply connected domain with smooth boundary $\partial \Omega$. Let $\Gamma_1$ be a subset of $\partial \Omega$ such that $\partial\Omega\subset\overline ...
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1answer
166 views

Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
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1answer
100 views

A pair of non-degenerate harmonic functions with orthogonal level curves

My problem is: Suppose $u$, $v$ are harmonic in region $\Omega$, and $\nabla u$, $\nabla v$ never vanish in $\Omega$. The level curves of $u$ and $v$ are perpendicular throughout $\Omega$. Moreover, ...
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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 ...
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1answer
209 views

Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
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1answer
35 views

How to construct a minimizing sequence?

Let ${u_k}$ be a harmonic function sequence that is continuous on a unit disk. How to construct the sequence such that, $ {u_k}$ are piecewise smooth and $u_k=0 $ on the boundary $ {u_k}$ make the ...
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1answer
213 views

Harmonic function vanishes with its normal derivative on a part of boundary; can Green's formula be applied to broken boundary? [duplicate]

Let $\Omega$ be an open domain, and let $\Sigma$ be a smooth and nonempty portion of the boundary. Let $u$ be a harmonic function in $\Omega$ and $u=D_\nu u=0$ on $\Sigma$. ($D_\nu$ is the derivative ...
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27 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 ...
2
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1answer
146 views

Harmonic function (PDE) - Orthogonal matrix

Let $u\in C^2(\mathbb{R}^n)$ be harmonic in $\mathbb{R}^n$, i.e. $$ \Delta u:=\sum\limits_{k=1}^{n}\frac{\partial^2 u}{\partial x_k^2}=0\mbox{ in }\mathbb{R}^n. $$ Let ...