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Questions tagged [potential-theory]

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

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Newtonian potential expansion identity

Preliminaries Consider the Newtonian potential $$\frac{1}{|\vec x - \vec y|}$$ with $\vec{x}, \vec{y} \in \mathbb{R}^3$ and $|\vec{x}| = x > y = |\vec{y}|$. Its Taylor expansion is given by $$\...
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Help with the specific improper integral for finding galaxy potential

I'm working on finding the potential of a razor-thin disk for a galaxy. I have encountered a specific improper integral for which I'm looking for an analytical expression. The integral is given below: ...
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Potential for Monotone Operator

I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The authors claim to construct a convex ...
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Real analytic potential 2

Let $G(x,\zeta)=|x-\zeta|^{2-n}$, for $n\geq3$, and $$p(x)=\int_{B(a,R )}G(x,\zeta)\Delta u(\zeta)d\zeta$$ with $\Delta$ meaning the laplacian, and $B(a,R )$ the ball of center $a$ and radius $R>0$...
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Upper envelope of plurisubharmonic functions

Suppose that $\{u_\alpha\}_{\alpha \in A}$ is a family of plurisubharmonic functions (psh function) on $\Omega \subset\subset \mathbb{C}^n$. Then, let $u(z) = \sup_{\alpha \in A} u_{\alpha}(z)$ be the ...
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Let $v(x) = \int_{B(0,R)} \frac{c}{||x-y||^{n-2}}\mathrm{d}y$. Show that for $x \in B(0,R)^c$ we have $v(x) = c_1||x||^{2-n} + c_0$

Let $$v(x) = \int_{B(0,R)} \frac{c}{||x-y||^{n-2}}\mathrm{d}y$$ Show that for $x \in B(0,R)^c$ we have $$v(x) = c_1||x||^{2-n} + c_0$$ Where $B(0,R) \subset \mathbb{R}^n$ is the ball centered at $0$ ...
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Give a subharmonic function that is not real analytic

Is there a function $f(x)$ defined on an open set of $\mathbb{R}^{2}$ ($ k\geq2 $) such that 1) $f$ is $C^{2}$ smooth, 2) $f$ is subharmonic, i.e. the laplacian $\Delta f$ of $f$ is positive, 3) ...
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Clebsch potentials vectorials manipulations

I have the Euler equation, by fluid dynamics: \begin{equation} \rho \left[ \partial_t \mathbf{v} + \left( \mathbf{v}\cdot \boldsymbol{\nabla}\right) \mathbf{v} \right] = - \boldsymbol{\nabla} p \...
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Real analytic potential

Let $B$ be an open ball in $\mathbb{R}^{k}$ ($ k>2 $) , and $\mu$ a Borel measure on $B$. We define $$p(x)=\int_{B}|x-t|^{-\alpha}d\mu(t)$$ on $B$, with $\alpha>0$ and $|.|$ the Euclidean norm....
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Definition of $C^{m,k/2}$-capacity of a point.

I hve come across the following notation and a new term $C^{m,k/2}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.
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Evaluate $\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx\text{ ,with }a>0$ [duplicate]

$$\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx,\;\;\;\;\text{with }a>0$$ How to evaluate Integral of $\ln(1+a^2-2a\cos x) dx$? where $x$ from $0$ to $2\pi$ and $a>0$, $\ln$ is the natural logarithm.
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When does a function on the circle have a finite Dirichlet energy extension to the disk?

It is often stated that the Dirichlet problem on a domain (say a disk in the plane) can be solved by finding the extension of the boundary data with minimal Dirichlet energy. However, this obviously ...
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Morrey embedding for Potential Spaces

I'm trying to prove that for $f\in H^{s,p}:=\{f/f=G_s*g,\,\,g\in L^p\}$ where $G_s$ is the bessel potential: $G_s(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}(1+|w|^2)^{-s/2}e^{iw\cdot x}dw$ the following ...
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Definition of exterior boundary of compact set

In potential theory, there is a result which states, Support of equilibrium measure of compact set belongs to exterior boundary of the compact set. But what is the definition of exterior boundary ...
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Charge density of charged conductor with flat side

Given a charged conducting body with a flat side, can the charge density (and hence the normal electric field) be constant on the flat part? According to physics lore, the charge density is greater ...
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60 views

Help with hyperbolic integral $\int_0^\infty x\frac{\cosh(bx)}{\sinh(x)}dx.$

The Integral Calculator couldn't help me with the following integral: $$\int_0^\infty x\frac{\cosh(bx)}{\sinh(x)}dx.$$ From some mathematical physics considerations, I get that the answer should ...
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How to calculate electric potential distribution of nodes in subgrid using FDM?

Boundary: the potential of the upper electrode is $7\,\text{V}$, and lower is $0\,\text{V}$. The potential of two edges is shown in the following figure: It is easy to calculate the potential ...
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Is there more to complex analysis than a two-dimensional potential theory?

Wikipedia entry on Potential theory in two dimensions says the following From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for ...
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Seeking an example of potential.

Consider a compact Lie group (the dimension of the group is greater than 1) and a representation $R^n$ of the group G. I am seeking an example of potential $p: R^n \rightarrow R$ which is invariant ...
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Example of discontinuous subharmonic function

Let $U$ be an open and connected subset of $\mathbb C$. $f:U \to [-\infty, \infty)$ is said to be subharmonic if its upper semi continuous and satisfy local mean value inequality. I want example of ...
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Question about general solution to Poisson's Equation

I've never taken a course on solving partial differential equations before, but I was wondering if my understanding on how to solve these types of questions is correct. Let's say I have an equation of ...
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Why is this a continuity principle?

I am trying to understand the following theorem from Ransford's Potential Theory in the Complex Plane intuitively. Why exactly is it called a continuity principle? Can someone explain? $p_{\mu}(z):= \...
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118 views

Understanding the proof of a continuity principle for potentials

I am reading Ransford's Potential Theory in the Complex Plane and I am stuggling with the inequality very last part of this proof (of (a)). Define $p_ {\mu} (z):=\int \log|z-w| \, d \mu (w)$. I ...
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Method of image charges for capacitor with fixed charge

If we have an infinite grounded plate at $z = 0$ and a point charge $q$ at some point $(0,0,a)$, then the method of image charges allows us to determine the electric potential by introducing a ...
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47 views

Use of monotone convergence on negative function?

I am reading a proof in Ransford's Potential Theory in the Complex Plane, where he uses the monotone convergence theorem on a negative function, and I do not understand why he can do that. It is ...
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Why is the velocity potential (in fluids) multi-valued in non-simply connected domains?

In Fluid Mechanics, you can define a velocity potential as a closed line integral in a vector field, which equals the sum of curl (vorticity) inside the line over which you are integrating (by Stokes' ...
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Newton Potential Estimates

Suppose that $ f: \mathbb{R}^n\rightarrow\mathbb{R} $ satisfies $|f(x)| \leqslant C <x>^{-k}$ for $k>2$ (Here $<x>^2 = 1+|x|^2$). And let $u(x) := \int_{\mathbb{R}^n} |x-y|^{2-n}f(y) ...
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How to predict the maxima and minima of a tidal wave?

Assume that a planet of mass $m$ and radius $r$ is orbiting a massive star of mass $M$ at a orbital distance $R$. Assume that the planet is covered in a ideal, non viscous ocean on a friction less ...
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Continuity of potential function at an interior point

Suppose $\Omega \subset \mathbb{R}^3$ is closed, bounded, with smooth boundary and the density $\rho: \Omega \to \mathbb{R}$ is continuous. The potential function is $$\phi(x) = \int_\Omega \frac{\...
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Differentiating the single-layer potential

Suppose $f\in L^2[-1,1]$ and consider the single layer potential with moment $f$ on $[-1,1]$ $$ Kf(x,y) = -\frac{1}{2\pi}\int_{-1}^1 \ln|(x,y) - (\xi,0)|f(\xi)\, d\xi $$ Formally I shown that for $x\...
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Potentials for Vector Fields on a Circle

This problem actually has three parts, and I have questions about each part. (What is in block quotes is the actual text of the problem.) Part (a): Consider the vector field on the circle given ...
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Functional inequality on $\mathbb{Z}^d$

Let $B_L = \{ x \in \mathbb{Z}^d : |x| < L\}$ be a box of side length $L \in \mathbb{N}$, where $| \cdot|$ is the $L_{\infty}$ norm and and assume $d \geq 3$. Let for any $L \in \mathbb{N}$ , $f_L :...
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Can we use double layer potential on semi bounded domain?

I would like to solve the problem of heat equation using double layer potential theory $$\frac{\partial W(u,x)}{\partial u}=\frac{\partial^{2} W(u,x)}{\partial x^{2}}$$ where $$\delta_{z_{0}}\left(x\...
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Show that the complex potential is $w(z)=k\ln(z)$

A line source of strength $k$ has velocity given (in cylindrical polars) by $v_{R}=\frac{k}{R}$, $v_{\theta}=0$. Show that the complex potential for such a source is $w(z)=k\ln(z)$. My solution so ...
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$W^{2,p}$ estimates for Newtonian potential

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f \in L^p(\Omega)$ for some $1<p<\infty$. Let $$ w(x) = \int_{\Omega} \Gamma(x-y)f(y)dy $$ be the Newtonian potential of $f$, ...
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Harmonic functions, equivalence of boundary conditions with phenomena outside domain.

I'm study a article and it contains: "It is well known that harmonic functions are associated with boundary conditions, which equivalently means that they depend upon the phenomena occurring ...
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53 views

logarithmic potential gives out a constant integral over an absolutely continuous measure

The question is as follows Prove that the integral $$\int_0^1\frac{\log|x-y|}{\sqrt{x(1-x)}} dx$$ is constant for $0<y<1$. This problem stems out from potential theory, and in fact it is ...
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From volume integral to surface integral in $n$-space

I'm stuck with this identity: $$\int_{|y|>r} f_y( \vec{y}+\vec{z} )y_i|\vec{y}|^{-n}d\vec{y}= \displaystyle \int_{|y|=r} -\frac{y_i^{2}}{r}|\vec{y}|^{-n}f(y+z)dS_x-\int_{|y|>r}f(y+z) \frac{\...
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1answer
109 views

Showing that a function is harmonic

Suppose that $\phi: \mathbb D \to \Omega $ is a conformal map from the unit circle to a Jordan domain $\Omega$. Fix a point $e ^{i \theta} \in \mathbb D $, how I can show that function $u: \mathbb D ...
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95 views

Green's function for upper half plane with pole at infinity

Can someone help me to calculate Green's function with pole at $\infty$ for the Laplace equation in the upper half plane domain $ \mathbb H = \{ (x,y) \in \mathbb R ^2 : y \geq 0 \}$. I want to ...
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Formula for equilibrium measure on [-1,1] for various kernels?

Do you know that what is the equilibrium measure on [-1,1] if the kernel is $|x-y|^s$? That is, for negative $s$ value I want to minimize the energy integral $$\int_{[-1,1]}\int_{[-1,1]}|x-y|^sf(x)f(...
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Clarification for definition of admissible: $\Delta\in (K)$

I am reading through the following book: E.M. Nikishin, V.N. Sorokin: Rational Approximations and Orthogonality, Translations of Mathematical Monographs, vol. 92, Amer. Math. Soc., Provindence RI, ...
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Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
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28 views

Finiteness of the Riesz potential of the Hausdorff measure restricted to a compact set

If $E$ is compact set in $\mathbb{R}^n$ with $0< H^s(E)<\infty$, where $H^s$ is $s$-dimensional Hausdorff measure and $0<s<n$, then fix $x$ and define $$ m(r):= f(B(x,r)) :=H^s(E\cap ...
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Dependence of $C$ in $\|\Phi\|_{H^{1}}+\|z\|_{H^{1}}\leq C\|u\|_{H(\operatorname{curl})}$

I am stuck at the following exercise: Let $\Omega \subset \mathbb{R}^2$. For $u \in H_{0}(\operatorname{curl},\Omega)$ there exists a decomposition $$u=\nabla \Phi +z$$ with $\Phi \in H_{0}^{1}$ and $...
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How to find the boundary between two source/sink distributions along two distinct curves seperated in space?

Consider a 2D Euclidean plane having two distinct sources distributions, each along the curves $S_1$ and $S_2$. The density of sources along both curves are identical and let it be equal to $m$. If I'...
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121 views

What domains can we give the Laplacian on the sphere $\mathbb{S}^2$ as an unbounded closed operator?

Recall from the theory of spherical harmonics on the sphere $\mathbb{S}^2$ that $L^2(\mathbb{S}^2)$ has an orthonormal basis of smooth eigenfunctions $Y_\ell^m$ for integer $\ell,m$ such that $-\ell \...
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An Estimate for Newtonian Potential

Consider the Newtonian Potential $$u(x)=\int_{\mathbb{R}^3} \frac{1}{|x-y|} f(y)dy$$, where $\mathbb{R}^3$ is the 3-dimensional Euclidian space. Show that if $|f(y)| \le C{|y|}^{-\alpha}$ for $2<\...
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333 views

Redistributing $k$ coins into $k$ stacks of height $1$

You are given k coins, arranged in k stacks of height 1. A “move” involves choosing a stack and redistributing all of its coins into other stacks, but when doing this each of the stacks can receive at ...
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Harmonic function radial on boundary

Let $u$ a harmonic function in $B_{1}(0)=\{x \in \mathbb{R}^{n}; |x| <1\}$, with $u(x)=v(r)$ in $\partial U$ then $u(x)=v(r)$, $\forall x \in B_1{(0)}$, i.e, a harmonic function radial on unit ...