Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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How to find numerically $\int_{0}^{1} w \langle \mathbf{p}, \mathbf{p}'\rangle \ du$ such $\nabla^2 w=0$ without solving the PDE for $w$ first?

Description Let $w(x, \ y)$ be a function that satisfies laplace's equation $$\nabla^2 w = 0 \ \ \ \ \ \ \text{on} \ \Omega$$ $$\dfrac{\partial w}{\partial n} = \langle \mathbf{p}, \ \mathbf{t}\...
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Electrostatic potential due to repeating coplanar charged strips (Mathews&Walker 5.1)

This problem is due to Mathews and Walker's Mathematical Methods of Physics, exercise 5.1. On the 2D plane, suppose we have a series of coplanar charged strips of line charge density $\lambda$ and ...
이희원's user avatar
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Questions about harmonic functions on Cayley graphs

I am reading A proof of Gromov’s theorem by Terence Tao, where I encountered harmonic (and Lipschitz) functions on Cayley graphs, here is the definition: Let $G$ be an infinite group generate by a ...
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Solving a Laplace Equation on a Semi infinite Strip

I am currently working semi-infinite strip question that requires the use of separation of variables. I inserted $q(x,y)=X(x)Y(y)$ into the Laplace equation provided in the image to then get $X''(x)Y(...
Jake Bynum's user avatar
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On the solution I got from 2D Laplace equation

$f$ is function that $f(0)=1$ and $f:[0,\infty)\to\mathbb{R}$. It's continuous on $[0,\infty)$ and is $C^2$ on $(0,\infty)$. Let $u$ be another function defined on $\mathbb{R}^2$ and $u(x,y)=f(\sqrt{x^...
Cunyi Nan's user avatar
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Prove that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$, $v$ being a solution to the given Poisson pde

Let $\Omega \subset \mathbb{R}^2$ and $v: \mathbb{R}^2 \to \mathbb{R}$ be a solution to: $$\begin{cases} v_{xx}+v_{yy} = -2 &\text{in $\Omega$}\\ v = 0 &\text{in $\partial \Omega$} \end{cases}$...
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How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?

I would like to see the calculations behind this proof I found here: Proof of polar coordinates theorem in Evans' PDE Book That is, I am struggling to understand the following: How do we ...
Silvinha's user avatar
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If $u\in W^{1,p}(\mathbb{R}^n) \cap C^1{(\mathbb{R}^n)}$ may I suppose that $|u(x)|=\frac{1}{\omega_n} \int_{B_{1}(x)}|u(x)|dy?$ How to prove it?

If $u\in W^{1,p}(\mathbb{R}^n) \cap C^1{(\mathbb{R}^n)}$ and $\omega_n$ is the volume of a ball, may I always suppose that $$|u(x)|=\frac{1}{\omega_n} \int_{B_{1}(x)}|u(x)|dy?$$ How to prove it? I ...
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Extending harmonic function to a codimension 2 subset.

Let $B$ be the unit ball. $K = \{(0,0,x_3,...,x_n)|x_3^2 + ... +x_n^2 \leq 1/2\}$. Let $u$ be a harmonic function on $B\setminus K$, then $u$ can be extended to a harmonic function in $B$. My attempt: ...
Alvis Zhalovsky's user avatar
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How to write the general solution to the laplace or helmholtz equation in 3D cylindrical coordinates

I have a finite cylinder with the solution satisfying the Laplace equation outside. Using separation of variables in cylindrical coordinates $(r, \phi, z)$ for the 3D Laplace equation, I get (with ...
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$-\log\delta_{\Omega}$ is plurisubharmonic.

I was reading the section of Pseudoconvexity and plurisubharmonicity of Hörmander's book (Introduction to complex analysis in several variables), and I have a doubt regarding the proof of the ...
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Solve Poisson problem $ \Delta u(\rho,\phi)=\ln \rho+2\cos^2\phi$ on annulus $1<\rho<3.$

Solve \begin{eqnarray} \left \{\begin {array}{lll} \Delta u(\rho,\phi)=\ln \rho+2\cos^2\phi&,~ 1<r<3,~0\leqslant \phi \leqslant \pi,~0\leqslant \phi < 2\pi\\ \displaystyle u_{\rho}(1,\phi)...
Nikolaos Skout's user avatar
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Question about Evans's proof of of the mean value forlmula for harmonic functions in Partial Differential Equations

I am trying to understand the proof of the mean value formula for Laplace's equation in Partial Differential Equations by Evans: If $u \space \epsilon \space C^2(U)$ is harmonic, then (16) is ...
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Find a and b so that a function is harmonic

The following exercise asks me to find $a$ and $b$ so that $v(x,y)$ is harmonic: $$ v(x,y) = y(bx^{2}-y^{2}) + \frac{y-a}{y^{2} + (x-a)^{2}} $$ So far I've tried both using Laplace (resulting ...
Pareja Facundo Jose's user avatar
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How to transform area integral $\int_{D} \omega^2 \ dx \ dy$ into boundary integral $\oint_{C} \square \ ds$?

Let $\omega$ be a function that satisfies the Laplace's equation $$\nabla^2 \omega = 0$$ The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the ...
Carlos Adir's user avatar
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Definition of the generic rate of a blow-up

I am reading a paper from van den Berg, Hulsof and King about blow-up solution for the harmonic map heat flow onto the sphere in a radially symmetric domain, this equation is given by, $$\theta_t = \...
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Equivalences of converse of mean value property, with $u$ not assumed harmonic.

Let $U\subset\mathbb{R}^n$ be open and $u\in C(U)$. Show the following statements are equivalent: $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}u(y)\,dy=u(x) $$ $$\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}...
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Strong maximum principle for a local maximum

Suppose $L$ is a strictly elliptic operator in the non-divergence form with $c \equiv 0$, $u \in C^2(\Omega)\cap C(\Omega)$ and $Lu \le 0$ in $\Omega$. Prove that if $u$ attains a local maximum at an ...
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How to show existence of harmonic functions on the unit disk

Hello StackExchange writers, I am currently working on Chapter 8 Problem 28 of Marsden/Tromba's Vector Calculus 5/E. The question asks for the following: Suppose $D$ is the disk $\{(x,y)|x^2+y^2 < ...
benhpark's user avatar
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What is the representation theory of the harmonic function $\frac{r}{x + i y}$?

Consider the function $f : \mathbb{R}^3 - (0,0,\mathbb{R} ) \to \mathbb{C}$ $$ f(x,y,z) = \frac{r}{x + i y} = \frac{\sqrt{x^2 + y^2 + z^2}}{x + i y}. $$ This function is harmonic, satisfying $$ (\...
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Understanding why the given function in harmonic

From L. Evans' book on PDE's: Definition The function $$\Phi(x) = \begin{cases} -\frac{1}{2\pi} \log|x| & (n=2) \\ \frac{1}{n(n-2)\alpha(n)} \frac{1}{\lvert x \rvert^{n-2}} & (n\geq3)\end{...
JackpotWizard 180's user avatar
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Understanding construction of fundamental solution to laplace's equation

We take a look at Laplace's equation $\Delta u=0, u:\mathbb{R}^n\rightarrow\mathbb{R}$ and want to look for explicit solutions, firstly, since Laplace's equation is invariant under rotations, for ...
JackpotWizard 180's user avatar
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Question on Theorem 2.2.13 in Partial Differential Equations by Evans

I'm working through the section on Laplace's Equation in Partial Differential Equations by Evans. I'm having trouble following a step in Evans's proof of the symmetry of Green's function (Theorem 2.2....
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How to solve the two-dimensional Laplace equation in this unbounded and multiply-connected domain?

I have the Laplace equation $$\nabla^2 \phi(x,y) = 0$$ where $\phi$ is a function defined on the domain $\mathbb{R^2}\setminus(C_1 \cup C_2)$. $C_1$ and $C_2$ are two circles of radius $r = 0.5$ and ...
Nicol's user avatar
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Solve $\nabla^2 u(x,y) = 1-y$ + B.C.

I am trying to solve: $$\nabla^2 u(x,y) = \partial_{xx} u(x,y) + \partial_{yy} u(x,y) = 1-y$$ over the domain $[-1,1]\times [0,1]$, with the B.C.: $$u(x,0) = 0 = u(\pm 1, y).$$ Is there any closed-...
anderstood's user avatar
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Question about Theorem 1 in Chapter 2 of Evans's Partial Differential Equations

I am working through Partial Differential Equations by Evans and I am struggling to understand a step of his proof of theorem 1 in chapter 2. He presents the following argument: The function $\phi$ ...
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poisson's formula for half-spce, evans page 38

I have questions about the proof of Theorem 14 (poisson's formula for half-space) in Page 38. Let $K(x,y)$ be the Poisson's kernel for $\mathbb R^n_+$: $$K(x,y)=\frac{2x_n}{na(n)}\frac{1}{|x-y|^n}\...
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Calculate the integral with a normalized kernel

$K$ is a baseline kernel function that is nonnegative, symmetric and supported on [-1,1]; $h$ is a bandwidth in local smoothing .Let $p_j$ denote the marginal density of $X_j$.We define the estimator ...
Lee's user avatar
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Does a gauge-invariant Caccioppoli inequality hold?

(I suspect that this question has an elementary resolution. But perhaps it would be more appropriate on MathOverflow, and if so I would not be opposed to migrating it there.) Let $V \Subset U$ be ...
Aidan Backus's user avatar
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1 answer
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Prove that $L^p\left ( \mathbb R^n, e^{\frac{-1}2 |x|^2} \right)\cap \theta(\mathbb R^n)$ is infinite dimensional

Prove that the space $$L^p\left ( \mathbb R^n, e^{\frac{-1}2 |x|^2} \right)\cap \theta(\mathbb R^n)$$ is infinite dimensional for $1\le p\le \infty$. Here, $\theta(\mathbb R^n)$ is the set of all ...
Sayan Dutta's user avatar
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Positive Harmonic functions that can be extended to boundary [closed]

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary $\partial \Omega$. Let $u \in C^\infty(\Omega)$ be a positive harmonic function. I wonder under which assumption, we can find ...
Ricci's user avatar
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Numerical method for calculating large discrete harmonic function

I am working on a problem, and for testing purposes, I want to do the following. Let be given a sparse graph $G(V,E)$ with edge weight matrix $W$, and $|V|$ large (anywhere from $10^3$ to $10^6$). We ...
blomp's user avatar
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1 vote
1 answer
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Laplace equation in an annulus

Suppose we fix $0 < R_{0} < R$ and $f: \mathbb{R}^{3} \to \mathbb{R}$ is the solution of the Laplace equation $-\Delta f = 0$, in the annulus $|x| \in (R_{0},R]$. Suppose $f$ is radial. Is it ...
InMathweTrust's user avatar
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Proving that any convex function on $\mathbb{R}$ is bounded above, then any subharmonic function bounded above on $\mathbb{C}$ is constant

The following question arises from another exercise of Ransford's book. Let $-\infty\leq a<b\leq+\infty$ and $f\colon(a,b)\rightarrow\mathbb{R}$. We say that $f$ is convex if for $$c=(1-\lambda)x+\...
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Representation formula for the $\Delta^2 u =f$

I already know the representation formula of $\Delta u =f$ is the that $u(x)$ equal to the $\phi * f(x)$, where $\phi$ is fandamental solution. Now I have wonder konw the representation formula of $\...
Apple Hsieh's user avatar
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Monotonicity of average in the border for sub/superharmonic functions, for non-euclidean balls

It is a well known fact that a harmonic function in $\mathbb{R}^n$ has the mean value property, namely: the average value of a harmonic function at the border of any (euclidean) ball is equal to the ...
confusedTurtle's user avatar
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1 answer
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weak version of maximum principle for not-quite-subharmonic functions

For smooth functions $f(x,y)$ in a disk, if $f$ is subharmonic then it satisfies the maximum principle. What happens if we relax the subharmonic condition, by requiring only certain bounds on $\Delta ...
Mikhail Katz's user avatar
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Laplace equation can implies some result

Prove that $$ \Delta u(x)=\Delta u(x_1,\ldots,x_n)=0 $$ also implies that $$ \Delta (|x|^{2-n}u(x/|x|^2))=0 $$ for $x/|x|^2$ in the domain of definition of $u$. I have no idea how to start with this ...
Apple Hsieh's user avatar
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exponential type and distribution of zeros

Let $H\doteqdot\{z\in\mathbb{C}\colon \operatorname{Re}z>0\}$ denotes the half plane and $f\colon H\rightarrow\mathbb{C}$ be a holomorphic function, with the property that $$f(n)=0\qquad\forall n\...
SprtWhitebeard's user avatar
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Upper bound on "time-continuous Zadoff–Chu signals"

I'm trying to find a tight (as tight as possible) upper bound for $\lvert x_u(t) \rvert$, where $x_u(t)$ is the $T$-periodic signal defined by the Fourier sum $$ x_u(t) = \frac{1}{N}\sum_{k=-N_0}^{N_0}...
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Convert Laplace's equation into polar coordinates

When I read Ahlfors's book, I came across this problem the$$u(x,y)$$is a harmonic function and satisfies Laplace's equation $$\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^...
tianhaowu's user avatar
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The identity theorem for harmonic functions

$\textbf{Background for question}$: Let $L_{k} = \{z \in \mathbb{C} : \arg(z) = \frac{\pi k}{n} \}$, and set $\Phi$ equal to the group of compositions of reflections in the lines $L_{k}, k=1,...,2n$. ...
porridgemathematics's user avatar
2 votes
1 answer
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A limit about a harmonic function

For $x = (x_1, x_2) \in \mathbb{R}^2$, let $$ E(x) = \frac{1}{2\pi} \log(|x|), $$ where $|x| = \sqrt{x_1^2 + x_2^2}$. In fact $E$ is the fundamental solution for the Laplace equation. Let $\Gamma$ be ...
Ayanamiprpr's user avatar
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1 answer
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Holder continuous for the Poisson integral to Laplace equation on Half space

From Evans, we see that if $u$ satisfies the Laplace equation on the half space with boundary $g$, $u$ is given by the Poisson integral: $$ u = \int_{\partial\mathbb{R}^n_+} \frac{2x_n}{n\alpha(n)}\...
Alvis Zhalovsky's user avatar
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Question in proof of Weyl's lemma on harmonic equations

I'm reading Jost's Partial Differential Equation Chapter 2. To point out where I'm stuck, here is the excerpt of the textbook dealing with Weyl's lemma. (The mollifier is defined as $\rho (t) = C exp(\...
Peter James's user avatar
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Reference to statement: A harmonic function on a compact connected riemannian manifold is constant.

I read this statement a few times, for example in the answer to this question. Does anyone have a reference to this statement?
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Harmonic L2-Forms

Let $(M,g)$ be a geodesically-complete Riemannian manifold. It is well-known that every harmonic $L^{2}$-function on $M$ is necessarily constant. The way to prove this is to show that every harmonic $...
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Solving an eliptical PDE in a rectangle

I was trying to solve this and got stuck right before the final solution, so that's the problem: $$ \begin{matrix} &u_{xx}+u_{yy}=0 & x\in (0,2), y\in (0,1) \\ &u(x,0)=0 & x\in[0,2]\...
Arie Creson's user avatar
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Product of two convex combinations is a harmonic function

Let two convex combinations $\lambda = \sum_{i = 1}^N \lambda_i \theta_i$ and $\mu = \sum_{i = 1}^N \mu_i \theta_i$ of real constants $\lambda_i$ and $\mu_i$, and variables $\theta_i$ such that $\sum_{...
Bridi's user avatar
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1 answer
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$\sum^{\infty}_{n=0}\frac{1}{2^n}\log |z-\frac{1}{2^n}|$ defines a subharmonic function over $\mathbb{C}$

I am preparing for my complex analysis qualifying exam. I encountered the following question that I don't really know how to solve. The question has two parts and I am able to solve the first part ...
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