For questions regarding harmonic functions.

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Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
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is this a harmonic function

Is this a harmonic function? $\frac{\ln1}{(x-2)^{2}+(y+1)^{2}}$ I think this is a harmonic function, because it's equal to zero ($ln1$). But maybe I'm wrong? Thanks!
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Harmonic function and Analytics

If both the real part and imaginary parts of an analytic function are harmonic. For this we have convers part? Can we prove harmonic function is analytic function?
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1answer
44 views

Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when ...
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29 views
+100

Variational formulation of harmonicity on Riemannian manifolds

$\newcommand{\R}{\mathbb{R}}$ I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps"). Here is the context: $M,N$ are ...
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1answer
19 views

Isometries of Riemannian manifolds are harmonic?

Let $(M,g),(N,h)$ be two Riemannian manifolds. Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional) (I know this is true when ...
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20 views

Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
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How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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Poisson integral formula the following Harnack inequality and Liouville theorem

Suppose $u$ is a nonnegative harmonic function in $B_R(x_0)\subset \mathbb{R}^n$. Prove by the Poisson integral formula the following Harnack inequality: $$ ...
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1answer
29 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
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25 views

Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
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Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
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35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
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Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
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34 views

A version of Casorati-Weierstrass for harmonic functions?

Suppose that $f:B(0,1)\setminus\left\{0\right\}\subset \mathbb{R}^n \to \mathbb{R}$ is a harmonic function. Clearly, the property that $\overline{f(B(0,\epsilon))}=\mathbb{R}$ for all $\epsilon>0$ ...
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Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
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Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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1answer
43 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset ...
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34 views

Harmonic Conjugate (Multiplication, power, addition/subtraction)

The Question Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic. I'm trying to prove that this function is also harmonic when $v$ ...
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Value of $f_x g_x+ f_y g_y + f_z g_z$ when $f$ and $g$ are harmonic

Let $f$ and $g$ be distinct real-valued harmonic functions, which not merely differ by a constant or are not merely multiples of each other. Also, assume that the first order partial derivatives of ...
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2answers
76 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
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1answer
42 views

Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = ...
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Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to ...
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1answer
30 views

Find all the harmonics functions constants on the rays

I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you. Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the ...
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19 views

Non trivial boundaries for laplacian equation on rectangle

1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically? $$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ ...
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1answer
29 views

Dirichlet problem to the ball with boundary data $1-2y^2$.

Let $\omega=\{(x,y):x^2+y^2<1\}$ be the open unit disk in $\mathbb R^2$ with the boundary $\delta\omega$.If $u(x,y)$ be the solution of Dirichlet problem $$\begin{cases} u_{xx}+ u_{yy}=0 & ...
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60 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
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43 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a ...
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Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
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1answer
48 views

The average of a subharmonic function on a circle increases with radius

Let $u$ be a subharmonic on open set $\Omega$. Let $a\in\Omega,R>0$ such that $B(a,r)\subset \Omega$. Prove $$v(\rho)=\int_0^{2\pi}u(a+\rho e^{it})dt$$ is a monotone increasing function on ...
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3answers
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Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
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1answer
29 views

Extending a harmonic function

Suppose that $u$ is a harmonic function on some open set $U$ (assume that $\overline{U}$ is compact). It is well known than in this case $u$ is smooth. Is it true that we can extend $u$ to the whole ...
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How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
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39 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...
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$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to ...
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Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
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Normal component of Laplace equation solution

Suppose $V\subset\mathbb{R}^3$ is a bounded region with a smooth boundary and that we are given $f:\partial V\rightarrow\mathbb R$. From classical PDE theory, the solution of the Laplace equation ...
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218 views

Is there a reason why Harmonic functions are defined on open sets?

Whenever I see a definition of a harmonic function, it's always defined as follows A function $f : U \to \Bbb{R}$ is called harmonic (where $U$ is an open subset of $\Bbb{R}^n$) iff it is twice ...
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1answer
32 views

The averages of a subharmonic function over concentric balls increase with radius

Let $B_r$ a ball of radius $r$ in $\mathbb{R}^n$ and $u \in H^{1}(B_r)$ with $\Delta u =0$ in the weak sense. I am reading a paper and the author says that : "since $|\nabla u |^2$ is subharmonic, ...
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Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
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1answer
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How to construct such a harmonic function on the upper half plane of $\mathbb{C}$ satisfying the following condition?

(1)Let u be a bounded harmonic function on the upper half plane of $\mathbb{C}$. Show that $\forall y$ we have $u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y\cdot u(t)}{(t-x)^2+y^2}dt$ for ...
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How to show that this integral is harmonic in both the upper and lower half plane?

The problem statement is: If $f(x) = \sum_j c_j \sin(a_jx)$ is a finite trigonometric sum given on the real axis $y=0$, show that: $$u(x,y) = ...
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Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or ...
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Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define ...
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Fluid Flow and Uniform Convergence of Taylor Series

I'm reading through a text on fluid flow and Laplace's equation, and it makes a statement that I do not understand and would really like to clarify. Here's the setup: Let $u:A \rightarrow R^2$ a ...
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I found a harmonic function from a convergent Laurent series; is this harmonic function unique?

I am guessing that it is simply, "yes", since the Laurent coefficients are unique. I solved for the coefficients to get the Laurent series, showed that it converges, and then took the real part of ...
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1answer
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Holomorphic versus harmonic functions

Is it true that any holomorphic function on domain $D$ is of the form $u_x-iu_y$ for some harmonic function $u$? Motivation for this question is the problem of existence of harmonic conjugates.
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69 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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Why does $v(z)=\text{Im}\left[\left(\frac{1+z}{1-z}\right)^2\right]$ not contradict maximum principle?

Since $\left(\frac{1+z}{1-z}\right)^2$ is holomorphic in $\mathbb{D}$, its imaginary part is harmonic, and we have $$\underset{r \uparrow 1}{\lim}v(re^{i\theta})=0 \quad \forall ~ \theta \in [0, ...
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What is an example of a nonconstant subharmonic function that attains a minimum?

Let $D$ be a domain in $\mathbb{C}$. What would be an example of a nonconstant subharmonic function that attains its minimum in the domain?