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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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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40 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
114 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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200 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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392 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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126 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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59 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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74 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
80 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
225 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 $\partial U^+ \cap \{x_n=0\}$. ...
2
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126 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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65 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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2answers
66 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
106 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
56 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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144 views

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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49 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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127 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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393 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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80 views

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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155 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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255 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?

What is a harmonic function? How do you prove that a function is a harmonic?
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1answer
55 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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48 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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67 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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147 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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110 views

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 ...
2
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1answer
90 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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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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48 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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108 views

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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83 views

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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62 views

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
121 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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97 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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79 views

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
156 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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34 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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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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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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121 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 ...
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1answer
44 views

Hausdorff dimension of support of harmonic measure in complex plane

I know that harmonic measure $\omega$ in complex plane $\mathbb{C}$ is absolutely continuous with Hausdorff measure $\mathcal{H_{h_k}}$ $(\omega << \mathcal{H_{h_k}})$, where $$ h_k(t) = t ...
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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 ...
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47 views

Prove that $u \circ f $ is plurisubharmonic on $\Omega_1$

I'm trying to show that the theorem in my book: Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on ...
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27 views

Prove that $\int_{{B(0,\epsilon)}\setminus \{z_1=0\}}\det\left(\text{Hessian}_u(z)\right)\mathrm{d}V=\infty$

I have a problem: For $u(z_1,z_2)=\left (-\log\left | z_1 \right | \right )^\alpha\cdot \left ( \left | z_2 \right |^2-1 \right )$, where $\alpha \in \left (0,1 \right )$. Prove that if $\alpha ...
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113 views

A family of curves orthogonal to another family. How to graph this?

I want to find a family of curves which are orthogonal to the family $$u(x,y) = e^{-x} \cos y + xy = const.$$ To this end, I check that $u$ satisfies Laplace's equation and so it has a harmonic ...
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34 views

Show that a $C^2$-function $u$ is plurisubharmonic if and only if the Hessian matrix $H_u(z)(\omega, \omega)>0$

I'm trying to show that the theorem in my book: A $C^2$-function $u$ is plurisubharmonic if and only if the matrix (the complex Hessian) $$H_u(z)=\left( \dfrac{\partial^2 u}{\partial z_j ...
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109 views

Green first identity and harmonic function

I proved the first Green identity $$\int_{\partial\Omega}f\cdot(D_ng)d\partial\Omega=\int_\Omega \bigtriangledown f \cdot \bigtriangledown g + f\bigtriangledown ^2g d\Omega$$ and now I need to prove ...