Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Bound by limit plane of harmonic function on half space vanishing at infinity

The problem is Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies \begin{cases} \Delta u(x,y,z)=0\\ \lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\ ...
vegetabledoge's user avatar
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The time-derivative of the Hamiltonian for a 1D harmonic potential

I do not undersand how to take the time derivative of the following Hamiltonian $\hat{H}(t) = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2(\hat{x}-a(t))^2$ where $a(t) = v_0t$. For instance how does $\...
Rillard's user avatar
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$L^1$ norm of spherical/circular Dirichlet kernel

I'm currently studying a particular Fourier multiplier and I came across the following question. The cubic $d$-dimensional Dirichlet kernel is \begin{equation} D_n(x)=\prod_{i=1}^d D_n^1(x_i), \end{...
Francesco_Trig's user avatar
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1 answer
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Algebraic approach to spherical harmonics

I am interested in an algebraic approach to the following theorem: Theorem. Consider the sphere $S^{n-1} \subseteq \Bbb{R}^n$ and for each $k=0,1,2,\ldots$, the space $H^k$ consisting of homogeneous ...
Michał Miśkiewicz's user avatar
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Entire function bounded in every horizontal line, and has limit along the positive real line

Let $f(z)$ be an entire function (holomorphic function on $\mathbb{C}$) satifying the following condition: $$|f(z)|\leq \max (e^{\text{im}(z)},1 ),\ \forall z\in\mathbb{C}$$ $$\lim_{\mathbb{R}\ni t\...
likeeatingoctopus's user avatar
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Comformal mappings of boundaries conditions

I'm reading the book "complex variables Demystified" from David McMahon and right now I'm in the chapter 12 where he explains how to solve Boundary values problems using comformal mappings ...
Juan Sin Tierra's user avatar
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Clarification question on applying divergence theorem to $\nabla \cdot (u \nabla u)$ on a compact manifold without boundary

I'm following the solutions given in this post. Basically, they are trying to prove that the kernel of the Laplace operator on a compact manifold without boundary is just made up of constant functions....
k12345's user avatar
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Computing value of a bounded harmonic function given outer limits

Problem: Let $u(z)$ be a bounded harmonic function in $\mathbb{D} = \{ |z| < 1 \}$ such that we have the limits $$\lim\limits_{r\rightarrow 1^-}u(re^{i\theta}) = \begin{cases}1 & \text{if } 0 &...
mathlover314's user avatar
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Continuity of subharmonic functions

I'm learning Perron's method and have a question on the definition of subharmonic functions. Definition Let $\Omega\subset \mathbb{C}$ be a bounded domain. A function u: $\Omega \to [-\infty, +\infty)$...
gaoqiang's user avatar
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Finding all harmonic functions on $n$-torus

Let $\mathbb{T}^n:=(\mathbb{R}/\mathbb{Z})^n$ be the $n$-torus. Then, I am trying to find all solutions of the equation $\Delta f =0$ on $\mathbb{T}^n$. However, I cannot really see how to find "...
Keith's user avatar
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Rapidly decreasing harmonic functions

Let $B \subseteq \mathbb{R}^m$ be an open ball of radius $1$, centered at the origin of the coordinate system, and $u$ a smooth harmonic function on the complement of the ball $B$. Now, if one assumes ...
Ivica Smolić's user avatar
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Mean value theorem for harmonic functions.

Suppose $u$ is harmonic on the $B(0,r)$ and continuous on $\overline {B(0,r)}$ then $$u(0) = \frac {1} {2 \pi} \int_{0}^{2 \pi} u \left (re^{i \theta} \right )\ d\theta.$$ I know the result for any $0 ...
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Difficulty in showing that $u$ is harmonic.

Let $\mu$ be a complex regular Borel measure on $\mathbb T.$ Define a function $u : \mathbb D \longrightarrow \mathbb C$ by $$u \left (re^{i \theta} \right ) = \int_{\mathbb T} \frac {1 - r^2} {1 + r^...
Anacardium's user avatar
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Constructing $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ with a given zero level curve

Let $C:\mathbb{R}\rightarrow\mathbb{R}^2$ be a differentiable, simple, regular parametric curve with an unbounded image. I need to construct a differentiable at least almost everywhere function $f:\...
FabrizzioMuzz's user avatar
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Is there an "easy" proof of the existence of radial limits of functions in the hardy spaces $H^p$?

Let $HOL(\mathbb{D})$ be the analytic functions defined on the unit disk and for $1\leq p \leq \infty$, $$H^p = \{f\in HOL(\mathbb{D}) : \lim_{r\nearrow 1} \int_{rC}|f(z)|^p \frac{dz}{2\pi i} < \...
travelingbones's user avatar
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Why the harmonic lifting of $v_n$ still holds $V_n(y)\to u(y)$

In the book "Elliptic Partial Differential Equations of Second Order", the theorem 2.12 says the function $f(x)=\sup_{v\in S_\varphi}v(x)$ is harmonic in $\Omega$. In the proof, it says ...
xyz's user avatar
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Understanding proof of Mean Value property for Harmonic Functions

In the proof of the mean value property for Harmonic functions, one often sees steps such as $$ E(r) := \frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)} u \ dy = \frac{1}{|\partial B(0,1)|}\int_{\...
kwerty's user avatar
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Arnold's Trivium problem 69

Does anyone have solution for Arnold's Trivium problem #69? Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside of the contour.
Ketchounez's user avatar
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Upper semi continuous regularization and Integral Means

Let $D$ be a domain and $u\colon D\rightarrow[-\infty,+\infty)$ be a locally integrable, locally bounded above function. Let us suppose that $u$ satisfies the local sub-mean inequality, that is for ...
SprtWhitebeard's user avatar
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Determine whether a function is the squared gradient of a harmonic function

Today I learned that, if $w$ (a real function defined on a region of $\mathbb R^n$) is a harmonic function, then $U:=\left|\nabla u\right|^2$ is subharmonic. This result makes me wonder whether there ...
Ulysses Zhan's user avatar
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Quick question about boundary conditions at the corners of the boundary on a rectangular domain

So as you can immediately see, I have 0 at my corners where I should instead have $u(0,H) = u(L,0) = 1$ and $u(L,H) = 2$. My problem was Laplace's Equation such that: $$u_{xx} + u_{yy} = 0 \\ \cases{u(...
Researcher R's user avatar
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Hypothesis on a parameter to ensure the unique solvability of a modified Laplace equation

Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following ...
SAKLY's user avatar
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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}\...
Carlos Adir's user avatar
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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 ...
Quzs's user avatar
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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}$...
nicoyanovsky's user avatar
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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 ...
Silvinha's user avatar
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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 ...
user911fas's user avatar
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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 ...
user 987's user avatar
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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 ...
TeaDrinker7's user avatar
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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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1 vote
1 answer
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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 = \...
Falcon's user avatar
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2 votes
1 answer
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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)}...
JackpotWizard 180's user avatar
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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 ...
M. Rubick's user avatar
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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 $$ (\...
user1379857's user avatar
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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
1 vote
1 answer
109 views

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
1 vote
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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....
TeaDrinker7's user avatar
3 votes
2 answers
157 views

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 ...
Adrian's user avatar
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2 votes
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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 $\phi$ ...
TeaDrinker7's user avatar
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1 answer
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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}\...
topst's user avatar
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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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