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Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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monotonicity of Dirichlet energy under different Dirichlet boundary conditions.

Let $\Omega$ be a domain with two smooth boundaries in $\mathbb{R}^n$. Suppose $u_1$ is a harmonic function in $\Omega$ satisfying that $u=0$ in $\partial_1\Omega$ and $u=1$ in $\partial_2\Omega$. ...
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taylor polynomial of a harmonic function is harmonic

Let $u \in C^2(\mathbb{R}^n)$ be a harmonic function and $m \in \mathbb{N}$. Prove that the Taylor polynomial $$ T_m(x) = \sum_{\alpha \in \mathbb{N}^n, |\alpha| \leq m} \frac{\partial^\alpha u(0)}{\...
hteica's user avatar
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Can a function that's second derivative zero have a local non global maximum or min?

Is the laplacian of a scalar function in 1D equal to the second derivative of that function? Can a Laplace equation only have a global maximum or minimum, as the value of the function is equal to the ...
dareen's user avatar
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Harmonic Conjugate of $\log|z|$ on Annulus? [closed]

Does $\log|z|$ possesses any harmonic conjugate in the annulus $B(0,r,1)=\{z:r<|z|<1\}$? Or equivalently I want to know if there is any holomorphic branch of logarithm that exists on the ...
Ravi's user avatar
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Question about Evans’ derivation of a Green's function

At page 34 of "Partial Differential Equations" by Evans, in order to define the Green function for the set $U$, the author defines a family of functions as the solutions of the boundary ...
Lorenzo Vanni's user avatar
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How to Derive the Poisson Kernel in Higher Dimensions?

I am trying to derive the Poisson kernel $ P(x, y) $ in higher dimensions, specifically in $ \mathbb{R}^n $. I know that the result should be: $$ P(x, y) = C_{n,a} \frac{y^{1-a}}{( |x|^2 + y^2 )^{\...
Christy's user avatar
2 votes
1 answer
50 views

When the average value of $f(x,y)$ over a circle of radius $R$ centered at $(x_0,y_0)$ differs from $f(x_0,y_0)$ by $\frac{R^2}{4}\nabla^2 f(x_0,y_0)$

The following definite integral equals the RHS for certain bivariate real-valued scalar functions $f$: $$\frac{1}{2\pi R}\int_{0}^{2\pi R}f\left(x_{0}+R\cos(\frac{s}{R}),y_{0}+R\sin(\frac{s}{R})\right)...
Simon M's user avatar
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3 votes
1 answer
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Harmonic function in half-disk

I'm studying for qualifying exam and was struggling with this problem. Thanks for any help! Let $\Omega$ be the half-disk $\{ z = x+iy : x > 0, |z| < 1\}$. Let $u$ be the bounded harmonic ...
Turtle5's user avatar
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Doubt in Hopf's Lemma

This is a problem from Evans' PDE book: Assume $u$ is connected. Use (a) energy methods and (b) the maximum principle to show that the only smooth solutions of the Neumann boundary-value problem: $$\...
Kadmos's user avatar
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Smooth real-valued harmonic function can be written as sum of real part and scaled log. [duplicate]

I am working on a previous year's qualifying exam problem, and I'm stuck. Here's the problem and what I know so far: The Question: Write $z=x+iy$. Let $f(x,y)$ be a smooth, real-valued, harmonic ...
qualsqualsquals's user avatar
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Sum of Dirichlet kernel for angle differences over $n$ angles on unit circle

Let $$D(\theta_i-\theta_j):= \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$ being the Dirichlet type of kernel of angle difference between $\theta_i$ and $\theta_j$ where $...
chloe's user avatar
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1 vote
1 answer
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Stein and Shakarchi Complex Analysis Chapter 8, Proof of Lemma 1.3

I am confused about the proof to Lemma 1.3 in Stein and Shakarchi's Complex Analysis Chapter 8 (Conformal Mappings). Here is a screenshot of the Lemma and its proof: My question is is the following: ...
Mashe Burnedead's user avatar
2 votes
2 answers
90 views

How to solve this Poisson's equation easily?

Here is the system of equations : \begin{cases} \Delta u=x^2y,&x^2+y^2 < a^2;\\ u=0,&x^2+y^2=a^2 \end{cases} I tried to let $v=u-\frac{x^4y}{12}$ , and I get \begin{cases} \Delta v = \Delta ...
cute dunkey's user avatar
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What boundary for Laplace's equation on a square gives the roughest interior?

Consider Laplace's_equation on an $N \times N$ grid of squares with Dirichlet boundary conditions boundary $B \to$ interior $X,\ (4N - 4)$ boundary points $ \to (N-2)^2 $ interior points. As a ...
denis's user avatar
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How can I find the argument of this complex numbers under the preimage of $f(\omega)=\frac{\omega+z}{1+\overline{z}\omega}$? [closed]

I have the following problem. I have the harmonic measure $\mu_D(z,d\omega)$ and I want to compute it on the unit circle i.e. $D=\mathbb{D}$. I already have shown that for any conformal map $f$, $\mu_{...
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Solving Laplace equation using Schwarz reflection

I'm studying Laplace PDE's and my teacher left as an exercise solving to following problem using Schwarz Reflection method: $\begin{cases} \Delta u = sen(u), \text{ in } B^+ \\ u(x,0) = 0, -1<x<...
76890's user avatar
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Estimate of Harmonic function

Let $f$ be harmonic function in $ \mathbb{R} \times ( 0 ,\infty)$. That is $ \Delta f = 0 $ in this domain. Then is the following true? $$ \frac{1}{r} \int_{r/2}^{r} \vert f(x,t) \vert dt \leq C \vert ...
User091099's user avatar
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1 answer
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Vector space of homogeneous harmonic polynomials

We know that the vector space of homogeneous harmonic polynomials of degree $l$ in $\mathbb{R}^3$ is of dimension $\dim(\mathscr{H}^{l}_3)=2l+1$. My professor gave an example of a basis in this space: ...
Krum Kutsarov's user avatar
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27 views

The function $u$ reaches its maximum on the boundary [duplicate]

my problem is: Let $\Omega \subset \mathbb{R}^2$ be a bounded domain and $v \in C^2(\Omega) \cap C^1(\overline{\Omega})$ be a solution of $$ v_{xx}+v_{yy}=-2 \quad \text{in} \quad \Omega $$ $$ v = 0 \...
the topological beast's user avatar
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Uniqueness of the exterior Neumann problem

I have a question regarding the uniqueness of the exterior Neumann problem, as stated by Lemma 2.4 in this paper. Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with $C^2$ boundary $\partial\...
Fluid's user avatar
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1 answer
69 views

Harmonic function in a square

I'm trying to find harmonic function $u(x,y)$ in a square $\Omega$ = {$ 0<x< \pi, \; 0 < y < \pi $} with boundary conditions $u(0, y) = u(\pi, y) = 0, u(x,0) =0, u(x, \pi) = 2 \sin(x) - \...
Malum Phobos's user avatar
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Dirichlet Problem with $L^p$ Boundary Data

I am seeking a proof of the following result related to the Dirichlet problem with $L^p$ boundary data. I am not quite sure how to approach the proof. Does anyone know where I might find such a proof ...
RiXaTorAgu's user avatar
1 vote
1 answer
54 views

A formula resembling the integral mean value on Kähler manifolds

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem: Theorem. ...
HeroZhang001's user avatar
  • 2,674
6 votes
1 answer
347 views

Laplace's Equation on a Pac-Man

I am struggling way too much with this problem, any help is highly appreciated. Consider the Pac-Man-like set described by $$P=\left\{(\rho\cos\theta,\rho\sin\theta):\rho\in(0,1),\theta\in\left(\frac{\...
tripaloski's user avatar
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Second Order Differential Equation- Underdamped Spring-Mass System [duplicate]

I have a spring with a some mass, m oscillating up and down, hooked to a ceiling with a circular disc attached to the bottom. Using newtons second law, $F = F_d + F_s$, where $F=ma$, $F_s = -kx$ (...
Eshwar Kolli's user avatar
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Isoperimetric inequality - references

Let $u$ be a harmonic function and $\Omega$ be a convex domain. Then is the following true? $$ \frac{\int_{\Omega} |\nabla u|^2 \,dx}{\int_{\partial \Omega} |\nabla u|^2 \,d\sigma} \leq \frac{\int_{B} ...
Adi's user avatar
  • 247
3 votes
1 answer
45 views

Is every bounded complex valued harmonic function on the open unit disc a sum of bounded holomorphic function and bounded antiholomorphic function?

Let $\varphi$ be a bounded complex valued harmonic function on the open unit disc. Then $\varphi = \psi + \bar \chi$ for some holomorphic functions $\psi, \chi$ on the open unit disc (see last line of ...
TheAlgebraist's user avatar
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48 views

What happens if constant appeared in seperating variables of cylindrical Laplace's equation is negative?

I'm trying to solve a Dirichlet problem of Laplace's equation in a region with cylindrical symmetry, i.e. \begin{equation} \begin{cases} \frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\...
ununhappy's user avatar
4 votes
1 answer
106 views

Doubt about a proof in complex analysis

In proving that a continuous function $$f:S(0,1)\to \mathbb{R}$$ Has a continuous extension to $$h:\overline{B(0,1)}\to \mathbb{R}$$ Such that $h|_{B(0,1)}$ is harmonic, one needs to show that the ...
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Product of Laplace-Beltrami eigenfunctions

Is there a simple proof (or counterexample) of the following: for a compact (Riemannian) symmetric space for which $\Delta f = f$ has no Nonzero solution, product of any two nonzero Laplace Beltrami ...
Areon's user avatar
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2 votes
4 answers
126 views

How to solve this 2D Laplace equation $-\Delta u=0 $ with $u(x,y)=y^{2}$ on boundary

$$ \mbox{I want to solve}\quad\Delta u = 0\quad\mbox{with}\quad u(x,y) = y^{2} $$ on boundary: The region is a $2D$ ball centered at $0$ with radius $1$. I want to write it in polar coordinates but it ...
YuerCauchy's user avatar
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34 views

For an analytic function $f$ in a domain $D$, if $\log|f|$ is harmonic in a neighborhood of $\partial D$ then $f\in C(\bar D)$.

I have stuck in one place while reading a paper. If $\phi$ is an analytic function on $D$, where $D$ is bounded multiple connected domains in $\mathbb{C}$. Now given that $\log\lvert \phi\rvert$ is ...
Ravi's user avatar
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1 answer
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Harmonic function agreeing on the boundary with a subharmonic function - existence?

A subharmonic function $g$ on the annulus is given and $U$ is defined to be a harmonic function agreeing on both boundary circles with $g$. I was wondering how to prove the existence of $U$. I'd ...
mikasa's user avatar
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2 votes
0 answers
33 views

Using a cutoff to show an interior gradient estimate for a harmonic function

I am trying to follow a calculation shown in Chapter 1.4 of Han and Lin's Elliptic PDE textbook. Let $u$ be a harmonic function on the unit ball $B_1$ in $\mathbb{R}^n$, and let $\eta \in C^\infty_c (...
maxematician's user avatar
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Solving Dirichlet problem on the unit disc. Is it correct?

I would like to solve the Dirichlet problem in $\Omega = B(i,2)$ and with boundary function $\varphi(x+iy) = x^2y^2$. Attempt I first consider the conformal map $f(z) = \frac{z-i}{2}$, with inverse $f^...
Mths's user avatar
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1 vote
1 answer
82 views

Solving the Laplace equation on an infinite strip

I wish to solve $u_{xx} + u_{yy} = 0$ on the domain $(x,y) \in \mathbb R \times (0,a)$ subject to the boundary conditions $$\begin{cases} u(x,0) = 0 \\ u(x,c)= \frac{x}{x^2+c^2} - \frac{x}{x^2 + 9c^2}...
RDL's user avatar
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2 votes
0 answers
33 views

Using conformal mapping to solve Laplace equation

Given the circles $$C_1 : x^2 + y^2 = 1,\quad C_2 : 5x^2 - 4x + 5y^2 = 0$$ let $D$ be the finite region between $C_1$ and $C_2$. Using the conformal mapping $$w = \frac{z-2}{2z-1}$$ solve the problem $...
idk31909310's user avatar
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112 views

Möbius transformation and Poisson kernel

Let's say $T(x,y)$ measures the temperature in degrees Celsius at points $(x,y)$ of a region $A$: $$A= \{ z \in \mathbb{C}: \text{Re } z> 0 , |z-2| > 1\}$$ If I put some ice at air pressure at ...
Mths's user avatar
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1 vote
1 answer
77 views

Does every Holomorphic function satisfy Laplace's Equation?

Consider Holomorphic function $f(z)$ $$f(z) = f(x + iy) = u(x,y) + iv(x,y)$$ Every holomorphic function is the sum of harmonic functions $u(x,y)$, $v(x,y)$, and so it is also harmonic and solves ...
Timothy Pulliam's user avatar
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28 views

Properties of Harmonic Functions and Monotonicity

I have a general question surrounding certain harmonic functions. I was able to solve the Laplace equation $\Delta f$ = 0 in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
HtmlProg's user avatar
1 vote
1 answer
47 views

Harmonic functions on an irreducible finite Markov Chain are constant

On Levin-Peres-Wilmer's book Markov Chain and Mixing Times there's the following Lemma With $P$ the transition matrix of an homogeneous markov chain over a finite state space $\Omega$, Lemma 1.16 ...
nsmon93's user avatar
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0 answers
128 views

Properties of superharmonic functions on compact subsets

I'm currently trying to prove the following - Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$ and $u \in C^2(\Omega)$ is a superharmonic function in $\Omega$, namely $-\Delta u(x) \geq 0$ ...
Thomas Petit's user avatar
2 votes
0 answers
62 views

Laplace equation in-between two non-concentric spheres

We fix two spheres $S_1$ and $S_2$ (without interior) and suppose that $S_2$ is entirely inside $S_1$. For example, $S_1 = \{x^2 + y^2 + z^2 = 25\}$ and $S_2 = \{(x-1)^2 + y^2 + z^2 = 1\}$. How to ...
dnes's user avatar
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1 vote
0 answers
51 views

Question on harmonic functions in a unit disc D

My question is: Let $u = u(x, y)$ and $v(x, y)$ be two harmonic functions defined in the unit disc $D = \{(x, y) : x^2 + y^2 ≤ 1\}$. If max $u ≥$ max $v$ on the boundary $∂D$, then $u ≥ v$ in $D$. I ...
starry41's user avatar
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0 answers
22 views

Understanding the Harmonic Energy for Maps Between Riemann Surfaces?

I am trying to understanding this functional, the thing I got confused about is the meaning of term $u_z\overline{u}_{\overline{z}}+ \overline{u}_{z}u_{\overline{z}}$ ? Because harmonic map is almost ...
ToastaFish's user avatar
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0 answers
34 views

Laplace's equation on half disc with a B.C

Solve for the Laplace’s equation on a half disc \begin{cases} ∆v(r, θ) = 0\\ 0 ≤ r ≤ 1, 0 ≤ θ ≤ π \\ v(r, 0) = 0, v(r, π) = π, v(1, θ) = 0 \end{cases} I used separation of variables i.e, $v(r,θ) = \...
JAEMTO's user avatar
  • 695
4 votes
1 answer
168 views

Solving the Poisson equation

I am an undergraduate student, in this semester I am taking the course of partial differential equations. So reading about Poisson equation by Evan's classic book for pdes, i have some questions: ...
user155's user avatar
  • 41
2 votes
1 answer
36 views

Finding the harmonic conjugate of the function

$u(x,y)=e^{x^2-y^2}(e^y \cos(x-2xy)+ e^{-y} \cos(x+2xy))$ Solving the Cauchy-Riemann equations for this is not practical. Alternatively, I could try to express $u(x,y)$ in the form of the real or ...
Derewsnanu's user avatar
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1 answer
59 views

A mathematical statement about Laplace's equation from famous physicist Landau's book

In the book titled Classical Theory of Fields, by famous Physicist Landau, writes, From $\nabla^2\phi(x,y,z)=0$, ...it follows, in particular, that the potential $\phi$ of the electric field can ...
Solidification's user avatar
0 votes
1 answer
71 views

Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither of the form $ax+by+cz+d$ nor a constant?

(Following an earlier post here) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither of the form $ax+by+cz+d$ ($a,b,c,d\in\mathbb{R}$) ...
Solidification's user avatar

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