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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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In a elliptic coordinate system,is there any existing solution for Laplace equation?

In a cylindrical system,Bessel function plays a role. And In a spherical system ,Legendre polynomials works in Laplace equations. Is there any solution for Laplace equation in a elliptic system? ...
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Subharmonic function on punctured disk

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Suppose $u \colon \overline{\mathbb{D}} \setminus \{0\} \to [0,\infty)$ is continuous and subharmonic on $\mathbb{D} \setminus \{0\}$. Show that ...
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Quotient space of harmonic functions on punctured plane

Let $a_1,\ldots,a_n$ be $n$ distinct points in $\mathbb{C}$ and let $\Omega := \mathbb{C} \setminus \{a_1,\ldots,a_n\}$. Define $H(\Omega)$ to be the space of harmonic functions on $\Omega$ and $R(\...
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Maximize derivative of harmonic function on the unit disk

I am interested in maximizing the partial derivative $$\frac{\partial u}{\partial x}(0,0)$$ for $u \colon \mathbb{D} \to [0,1]$ a harmonic function defined on the unit disk $\mathbb{D} = \{(x,y) \in \...
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2answers
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Harmonic conjugate and resulting analytic function

I can't quite figure out the last part of this problem... Find a harmonic conjugate of the harmonic function $u(x,y)= x^3-3xy^2$. Write the resulting analytic function in terms of the complex ...
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Subharmonic function bounded by linear functions

Problem: Let $\Omega = \{(x,y) \in \mathbb{R}^2 : xy > 0\}$ and suppose $u \colon \overline{\Omega} \to \mathbb{R}$ is continuous and subharmonic on $\Omega$ and satisfies $$u(x,y) \leq \sqrt{x^2 + ...
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Laplace Equation in Rectangle: Boundary Value Problem [closed]

Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(...
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Max vs. min bound for nonnegative harmonic function

Problem: Let $\Omega$ be an open, bounded, simply connected subset of $\mathbb{C}$ and let $u \colon \Omega \to \mathbb{R}$ be a nonnegative harmonic function. Show that for each compact subset $K \...
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Rado–Kneser–Choquet Theorem proof

I was reading the proof of the Rado–Kneser–Choquet Theorem. The statement is there in the image (taken from the book "Harmonic Mapping in the Plane, Duren page-$30$": In the proof, he shows that ...
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Show explicitly that the gradient of two harmonic functions are orthogonal

So form a prior problem stating that $$u(x,y)=\sin x\cosh y\qquad (1)$$ I found the harmonic conjugate, analytic function and found out that it indeed was harmonic $(u(x,y))$ that is. $$f(z)=\sin z+...
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Linear harmonic functions

Let $u(x,y)$ be a harmonic function on $\mathbb{R^2}$ and $v(x,y)$ be a harmonic conjugate of $u(x,y)$ on $\mathbb{R^2}$. Suppose that the partial derivative $v_x(x,y)<C$ for a real constant $C$. ...
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If $u>0$ is a harmonic function on $\mathbb{R}^n$, show that there exists $a,b\geq 0$, s.t $u(x)=a+b|x|^{2-n}$,for all $x\in\mathbb{R}^n-\{0\}$

If $u$ is a positive harmonic function on $\mathbb{R}^n(n\geq2)$, how to show that there exists $a,b\geq 0$, such that $u(x)=a+b|x|^{2-n}$,for all $x\in\mathbb{R}^n-\{0\}$
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Laplace Equation with Inhomogeneous Boundary Condition

I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous. I'm having trouble deciding how to handle one of the boundary ...
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2D partial differential equation with boundary values (Dirichlet problem)

I'm trying to solve a differential equation (Dirichlet problem): $\Delta \varphi(x, y) = 0$ $0 \leq x \leq a, \ \ \ 0 \leq y \leq a$. The boundary values are: $\varphi(a, y) = \varphi(x, a) = \...
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harmonic function degenerating at infinity

If $\Omega\subset R^n$ is a bounded domain with smooth boundary. Prove that there is a positive harmonic function $u$ on the complement of $\overline{\Omega}$, with $u=1 $on $\partial\Omega$ and ...
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1answer
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How upper bound of harmonic sum is proven

I know it’s that kind of $1 + \log_{2}{n}$, I don’t know where did that come from etc...
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Partial Differential Equations: Dirichlet problem for the Laplace equation on half-space with boundary condition

Consider the Dirichlet problem for the Laplace equation on half-space with the boundary condition $f(x)=\chi _{[0,\infty)}(x)$. Use the exact solution and find a functon $g:\mathbb{R}\rightarrow \...
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69 views

Nonconstant solutions of homogeneous Poisson equation on domain without boundary

This answer on physics stack exchange says For a compact domain without boundary (such as the surface of a sphere), you don't need any boundary conditions: there are no non-constant harmonic ...
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2D Laplace Equation in Polar Coordinates on Half-Disk with Neumann Conditions

Assume the 2D Laplace Equation $$\nabla^2 u =0 $$ in polar coordinates $(r, \theta)$. The general solution is $$u(r,\theta) = \frac{A_0}{2} + \sum^{\infty}_{n=1}r^n[A_n\cos(n\theta) + B_n\sin(n\theta)...
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Green's function for $\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}$

Compute the Green's function for the Laplacian, for the region $$\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}.$$ My approach is to use a reflection argument similar to the one used for ...
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Uniqueness of the potential flow past a cylinder

I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \...
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Relation between Warping Function and Solution of Poisson's Equation

In the solution of torsion problem for non-circular cross-sections, the warping function is defined. Without going into the details, the torsion constant is defined as follows: $$ J = \int_A \left[ \...
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Solve this PDE using Fourier Transform

Question: Solve the following system using Fourier Transform: \begin{alignat}{2} & \nabla^2 f = 0 & z<0 \\ & \frac{\partial f}{\partial z} = \frac{\partial g}{\partial x} \quad \frac{\...
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If v is harmonic conjugate of u, then the harmonic conjugate of $3u^2 − v^3$ is harmonic conjugate of

Attempt: I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to $3u^2 − v^3$ using any other holomorphic functions,...
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1answer
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Does having harmonic functions $u(x,y)$ and $v(x,y)$ guarantee having an analytic function?

Verify that each given function $u$ is harmonic (in the region where it is defined) and then find a harmonic conjugate of $u$. (a) $u=y$ I was able to verify that $u$ is harmonic pretty easily. But ...
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1answer
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Finding the harmonic conjugate of a function

The problem: Find the harmonic conjugate of $G(x,y)= 2v^2(x,y)-2u^2(x,y)$ My attempt to solving it I know that "If two given functions u and v are harmonic in a domain D and their first-order ...
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1answer
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Find a and b values so a given function is harmonic

The Problem: Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ be an entire function. If $g(z)=au^2(x,y) - bv^2(x,y)$ find values for a and b so $g(z)$ is a harmonic function. My attempt to find a solution: Well ...
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Show that the equation defines a harmonic conjugate

Suppose $u$ is a twice continuously differentiable real-valued harmonic function on a disk $D(z_0;r)$ centered at $z_0 = x_0 +iy_0$. For $(x_1, y_1) \in D(z_0;r)$, show that the equation \begin{align*...
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An estimate for the conjugate Poisson Integral on the unit disk.

Let $f \in L^1(\Bbb{T})$ and $Q_r(\theta)=\frac{2r\sin{\theta}}{1-2r \cos{\theta}+r^2},r \in[0,1]$ the conjugate Poisson kernels and $P_r(\theta)=\frac{1-r^2}{1-2r \cos{\theta}+r^2},r \in[0,1]$ and $\...
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Conditional expectation of a bounded harmonic function

Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map ...
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find a harmonic function $v$ such that $v$ takes 1 on $L_1$, $3$ on $L_2$, and equals $5$ on $L_3$

Let $D=\{z: \Im z>0 \text{ and } |\Re z|<\pi/2\}$ be a half strip. Let $L_1$ be the left boundary $L_1:=\{z:\Re z=-\pi/2\ \text{ and } \Im z>0 \}$ and $L_2=\{z:\Re z=\pi/2\text{ and } \Im z&...
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The dirichlet and harmonic functions why they are important

I am wondering why finding a function that is harmonic on the sphere and that respect some conditions on the frontiere of the sphere is important ? This is called the Dirichlet problem, and I don't ...
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P-norm of the Poisson Kernel

If we have $P_r(\theta)=\sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta}=\frac{1-r^2}{1-2r\cos(\theta)+r^2},$ Is there a quick way to compute the norm $\|P_r\|_p$, or an upper estimate?
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Find a harmonic function satisfying the boundary conditions in the given domain

I'm confused with this question since there is no mapping given. I have a picture of the domain but I don't know how to find u(x,y). Where do I start?
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Exercise Jurgen Jost's PDE show that u harmonic and nonnegative is constant

2.5: Let u be harmonic and nonnegative, show that u is constant. (Hint use the previous exercise). The previous exercise was posted in another question, stated the following. 2.4: Let $u:B(0,R)\...
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Poisson problem, minimun value of subharmonic function in the interior.

Consider the problem: $$ -\Delta u = - f(x_1,x_2), \text{in } \Omega$$ $$ u = f(x_1,x_2), \text{in } \partial\Omega$$ Where $\Omega=B(0;1) \subset \mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$. Is it ...
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Exercise in Jurgen Jost: Show that if $u$ is harmonic then $v$ is harmonic

Let $\Omega \subset \mathbb{R^3-\{0\}}$ and $u:\Omega \rightarrow \mathbb{R}$ harmonic in $\Omega$. Show that $$v(x^1,x^2,x^3):=\dfrac{1}{|x|}u\Big(\dfrac{x^1}{|x|^2},\dfrac{x^2}{|x|^2},\dfrac{x^3}{|x|...
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Finding / creating harmonic functions with given criteria

Find a function $u$ harmonic on $\{Im(z)>0, Re(z)>0\}$ with boundary values $0$ on $\{Im(z)>0, Re(z)=0\}$ and 1 on $\{Im(z) = 0, Re(z) > 0 \}$. How does one go about this? My professor ...
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Generalize $\sum_{n=1}^{\infty}\frac{H_n^2-H_n^{(2)}}{(n+1)(n+2)}=2$

In this paper on section [5], Recently J. Choi [4, Corollary 3] proved a sequence of identities: $$\sum_{n=1}^{\infty}\frac{H_n^2-H_n^{(2)}}{(n+1)(n+2)}=2\tag1$$ Let just generalize $(1)$ $$\sum_{...
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finding Ker(T) of a parameter's linear transformation

I am suppose to find the ker(T) of linear transformation of: $$ G\begin{pmatrix}a & d \\ c & b\end{pmatrix}= a+\frac{b+c}{2}x+\frac{b-c}{2}x^2 $$ the form $T:V \to W$ My problem is that I ...
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Regularity for weak solution of Poisson problem in a rectangle

Let $\Omega=(0,1)^2$. Let $u$ be a weak solution of $\Delta u=f$ con $f \in L^2(\Omega)$ e $u \in H^1_0(\Omega)$. I would like to prove that $u \in H^2(\Omega)$. I know that $u \in H^2_{loc}(\Omega)$ ...
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1answer
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Excercise 2.4 in Jurgen Jost's PDE “Harnack's inequality” for harmonic functions defined on a ball

Let $u:B(0,R)\subset \mathbb{R^d}\rightarrow\mathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$\dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)\leq u(x)\leq \...
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Use of Holder inequality in gradient estimate for harmonic function.

While reading the book "Elliptic Partial Differential Equations" by Han and Lin, I failed to understand the proof of the interior gradient estimate for harmonic functions. The theorem says that if $u$ ...
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Is $\ln|z|$ harmonic in the punctured disk [closed]

1. how can i show that $\ln|z|$ is harmonic in punctured disk ? also $\ln|z|$ has no harmonic conjugate in $\Bbb C\setminus\{0 \}$ but has in $\Bbb C\setminus[0, \infty)$.
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1answer
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Solving the Dirichlet Problem for an infinite strip

I have been looking into the Dirichlet problem and conformal mappings, but am unsure as to how to find a solution $u(x, y)$ for the Dirichlet problem given this information: The region is $U = \{\ x+...
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1answer
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Are the singular points of harmonic function on the disk always of measure zero?

Let $f : \mathbb D^2 \to \mathbb R$ be a smooth function with no singular points, i.e. $df \neq 0$ on $\mathbb D^2$. (Here $\mathbb D^2$ is the closed unit disk in $\mathbb R^2$). Let $\omega:\mathbb ...
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2answers
44 views

Liouville's theorem for harmonic functions

I was reading the proof of Liouville's theorem for harmonic functions (in $\mathbb{R}^n$) in Wikipedia, but I could not understand where do they use in that proof the assumption that $f$ is bounded. ...
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26 views

Bound on gradient positive harmonic function

My question is essentially the same as An inequality concerning an harmonic function , however I did not find the answer given satisfactory. To restate it, I would like to solve the following: Let $h$...
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0answers
34 views

Mapping a curve-sided quadrilateral to a rectangle

I am currently investigating different ways of solving the Laplace equation $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial z^2} = 0 $$ numerically on the domain $\Omega$ shown as ...