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12 views

Is the potential of a periodic conservative field periodic?

Let $Y = [0,1]^3$ and consider a conservative vector field $F$. Denote its scalar potential by $\varphi$, i.e. $$ \nabla \varphi = F. $$ If $\varphi$ is $Y$-periodic it is clear that $F$ is periodic, ...
2
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0answers
18 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
1
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0answers
22 views

Newtonian potential of torus

Is there an explicit expression of the Newtonian potential for torus? As the expression for ball is clear by calculation. Definition of Newtonian potential of domain $\Omega$ at x is defined to be ...
0
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1answer
36 views

Complex potential between axes & hyperbola

okay so i have searched through the entire net and i only get what is complex potential theory. But no body explain how to solve the question maybe because that's at the basic level. My question is ...
0
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0answers
9 views

Energy functional of a uniform distribution of mass on a circle

Define $\nu$ as the measure in the Borel sets of $\mathbb{R}^{2}$obtained by uniformly distributing a unit mass on the circumference $\vert x \vert = 1$. Define its energy as $$I_{2}[\nu] = ...
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1answer
36 views

Show that requiring Electrostatic potential to be a stationary point of Electrostatic potential energy is equivalent to Laplace's equation.

Suppose we want to find the electrostatic potential $\phi$(r) inside of some volume $V$ with known boundary conditions. The physical field configuration should minimize the electrostatic potential ...
3
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0answers
60 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
0
votes
1answer
120 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
5
votes
2answers
98 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
1
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1answer
42 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
0
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1answer
28 views

What is $s$ in s-energy (eg. Riesz s-energy)

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find ...
0
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0answers
29 views

How to get between the equivalent definitions of Newtonian capacity?

Here are the two definitions: (1) $$\mathrm{Cap}(A)=\left[\inf \left\{\int\int |x-y|^{d-2}\mu(dx)\mu(dy):\mu \; \text{a probability measure on} \; A \right\}\right]^{-1}$$ and (2) ...
2
votes
1answer
26 views

Taylor expansion of the electrostatic potential $1/\|\cdot \|$

I have stumbled over this problem several times in electrodynamics, and I just don't get the hang of it. The task is to do a Taylor expansion of $\,f(\vec{x},\,\vec{a}) = ...
2
votes
1answer
41 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
0
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1answer
30 views

Dirichlet boundary value problem in convex domains with discontinuous boundary values

Consider $\Omega$ an open, bounded, and convex domain in $\mathbb{R}^n$. Let $g \in L^{2}(\partial \Omega)$ such that the problem $$ \left\{ \begin{array}{ccccccc} \Delta u = 0, \ \text{in} \ ...
8
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1answer
76 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
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0answers
28 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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4answers
64 views

Evaluation of the integral $\int_{0}^{R}\int_{-1}^{1} r^2/\sqrt{r^2+L^2+2L\alpha}\,d\alpha dr$

I am trying to solve this integral $$ \int_{0}^{R}\int_{-1}^{1}\frac{r^{2}\,{\rm d}\alpha\,{\rm d}r}{\, \sqrt{\vphantom{\Large A}\,r^{2} + L^{2} + 2L\alpha\,}\,} $$ where $L$ is some positive ...
0
votes
1answer
32 views

The limit function of decreasing sequence of subharmonic is also subharmonic

Let $u(z)$ be a continuous function on a domain $D \subset \mathbb{C}$ to $[−\infty, \infty)$. Suppose $u_n(z)$ is a decreasing sequence of subharmonic functions on $D$ such that $u_n(z) \to u(z)$ for ...
2
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1answer
44 views

Computing Newtonian capacity of sets like intervals, discs?

For a metric space $(E,\rho)$ the $a$-capacity is defined as $$\mathrm{Cap}_{a}(E)=\left[\inf\left\{\int \int \frac{d\mu(x) \, d\mu(y)}{\rho(x,y)^{a}}:\mu\text{ probability measures on ...
0
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1answer
35 views

If $f$ and $g$ are holomorphic, then $\log(|f|+|g|)$ is subharmonic

Let $f$ and $g$ be two holomorphic functions on a plane domain, and let $u(z)=\log(|f(z)|+|g(z)|)$. Is it true in general that $u$ is subharmonic? I know it is true if $g=0$, but here I have some ...
0
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1answer
50 views

Question about $\int_{-1}^{1} \log |z-x| dx, z \in \mathbb{C}$.

I am wondering if this integral can be evaluated using the "usual" integration techniques. I have written $| z - x | = \sqrt{(\operatorname{Re}({z}) - x)^2 + \operatorname{Im}(z)^2}$ but it doesn't ...
1
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1answer
20 views

Showing there's no potential for a vector field on some region of the space?

Imagining we have a vector field $f$ for which $\operatorname{curl} f = (0,0,0)$, why can there be a potential to it on some region and not on some other. e.g. what kind of reasoning can prove that ...
1
vote
2answers
57 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
0
votes
1answer
41 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
0
votes
1answer
28 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
votes
1answer
43 views

Some properties of capacity

Let $\Omega\subset\mathbb{R}^N$. For $K\subset \Omega$ we can define the $p$-capacity, $p\in (1,\infty)$ as the number $$\operatorname{cap}_p(K)=\inf \int_\Omega |\nabla u|^p$$ where the infimum is ...
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1answer
30 views

Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
1
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0answers
322 views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
0
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0answers
50 views

Help interpret Sommerfeld radiation condition.

I am studying the Sommerfeld radiation condition. In potential theory, a solution $u(r,\theta)$ to a partial differential (such as the Helmholtz equation $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
4
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1answer
75 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=\inf\int_E\int_E \log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $\inf$ is taken over ...
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1answer
291 views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
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0answers
42 views

Dirichlet boundary problem and volume potential

Consider the Dirichlet boundary value problem for the Laplace-equation (1) $-\Delta u=f$ in $\Omega:=B_R(0):=\left\{x\in\mathbb{R}^3: \lVert x\rVert < R\right\}$ (2) $u=0$ on ...
3
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0answers
365 views

Finding the Green Function of the upper half ball

Find the Green function of $\Omega:=\left\{x\in\mathbb{R}^n:\lVert x\rVert<R, x_n>0\right\}$ and show that the function you've found is indeed a Green function! You are allowed to use ...
2
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2answers
97 views

Assumption on $u$ harmonic in disc making $u$ unique with particular (discontinuous) boundary data

From an old qualifier: Let $$J_1 = \{e^{i\theta}: 0 < \theta < \frac\pi2\}, \,\,J_2 = \{e^{i\theta}:\frac\pi2<\theta< \pi\},\,\, J_3 = \{e^{i\theta}:\pi<\theta < 2\pi\}$$ be ...
1
vote
1answer
121 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
1
vote
2answers
92 views

Line integral over a vector field using a potential function.

Let $X(t)=\left ( \pi\cos t, t, e^{\sin t} \right ), 0\leq t\leq \pi $. And let $\textbf{F}(x,y,z)=\left ( 2xy+\frac{-y}{x^2+y^2}, x^2+ze^{yz}+\frac x{x^2+y^2}, ye^{yz} \right )$ be a vector field. I ...
0
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0answers
30 views

Two more little tasks concerning Dirichlet boundary value problem

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap ...
0
votes
1answer
36 views

Two little tasks concerning Dirichlet boundary value problem

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap ...
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0answers
124 views

Green function of the first quadrant

Find the Green function of the first quadrant $x_1>0, x_2>0$. HINT: Use the Green function of the half space $\Omega:=\left\{x\in\mathbb{R}^n : x_n > 0\right\}$ which is given by ...
2
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1answer
79 views

How would you prove $E = -\vec{\nabla} V$ from the electric potential's line integral?

Basically, we were given an equation: $$V_1 - V_2 = \int_{r_1}^{r_2}\vec{E}\cdot d\vec{r}$$ where $\vec{E}$ is the electric field distribution and $d\vec{r}$ is the displacement vector of the charge. ...
2
votes
1answer
128 views

Find the Green function of the upper half ball by using the Green function of the whole ball

Find the Green function of the upper half ball $\Omega:=\left\{x\in\mathbb{R}^n|\lVert x\rVert<R, x_n>0\right\}$ (for the Dirichlet boundary value problem of the Laplace equation). Show ...
0
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1answer
38 views

What is the consequence for the dyson sphere? [closed]

as you can see here Normal derivation the volume potential is constant in the origin and inside the hollow ball. What consequences does this have for a dyson-sphere? I do not know, because I ...
1
vote
1answer
152 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
0
votes
1answer
56 views

Volume potential (show continuity)

Consider $$ \Omega:=B_{R_2}(0)\setminus \overline{B_{R_1}}(0)\subset\mathbb{R}^3 $$ with $0<R_1<r_2<\infty$ (hollow ball) and $$ ...
1
vote
1answer
90 views

Show that the Kelvin transformation is a diffeomorphism and find the inverse

Let $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. We call $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ with $$ y=\phi(x):=\frac{R^2}{\lVert ...
5
votes
1answer
117 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
2
votes
1answer
57 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
2
votes
1answer
43 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
4
votes
1answer
270 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...