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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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Is there some link between analytic continuation and potential theory?

It seems like the analytic continuation of a function has a lot in common with the process of trying to define a potential for a vector field (or a differential 1-form). In particular, an analytic ...
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Inconsistency in the Fundamental Solution Constant for the Laplacian in Higher Dimensions

I am working through a problem involving the extension problem for fractional Laplacians, and I've encountered some inconsistencies in the derivation of the fundamental solution and the associated ...
PowerPoint Trenton's user avatar
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Complex Potential Between Axes and a Hyperbola

I'm looking at the exact same question as here: Complex potential between axes & hyperbola (Advanced Engineering Mathematics 8th edition, Erwin Kreyszig, problem 12 of section 16.1) Find the ...
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Why the log function is so important on the plane?

I am studying right now some Complex Analysis and I have seen the importance of the (complex) logarithm function in almost every subject in it. Now I'm intrigued with that (possible) relation between $...
underfilho's user avatar
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An inequality for continuity of normal derivative of double layer potential

Let $\Gamma \subset \mathbb{R}^2$ be a $C^2$ smooth Jordan curve, $t_x$,$n_x$ be the tangent and exterior normal of $x \in \Gamma$ respectively. $$ U(x,t_x,\delta,n_x,\delta) := \{ y = x + \xi t_x + \...
Yidong Luo's user avatar
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Potentials and orientation-preserving isometries

I'm trying to prove the following Lemma: If $U,V \subseteq \mathbb{C}$ are open, and are equipped with conformal metrics $g_U=\lambda^2dzd\bar{z}$ and $g_V=\mu^2dzd\bar{z}$. If $f:U \rightarrow V$ is ...
Lazarus Frost's user avatar
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discontinuity of double layer potential kernel on $\mathbb{R}^2 \times \Gamma$ and $\vert x- y \vert >0$

Background There is a result in [SV02] that Lemma 2.5.1 Assume that $\Gamma $ is a $C^1$ smooth Jordan curve. Suppose that $g(x,y)$ satifies (1) Function $g(x,y)$ is continuous in the set $\mathbb{R}^...
Yidong Luo's user avatar
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Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$

Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following: Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
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Question on a hydrodynamic double layer potential in Ladyzhenskaya's book

In O.A. Ladyzhenskaya's "mathematical theory of viscous incompressible flow" there appears the following integral potential $$ W_i(x)=\int_{S} K_{ij}(x,\eta)\phi_j(x)dS_\eta, K_{ij}(x,\eta)=-...
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Is there a Dirichlet form on product space?

Let $(X_t)_{t\ge 0}$ be a Markov process on a locally compact Polish space $E$ (say for example the Sierpinski Gasket). Then there is a Dirichlet form associated with $X_t$ on $E$, call it $({\cal E},{...
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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: ...
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Potential density of killed brownian by local time

Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(-...
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Complex potential/ observable

I have an observable denoted by $C$, related to a complex potential $B$ by : $$ C= \bar{B}B ,$$ where $B$ is a complex potential. I know that $ \left. C \right|_0 =C_0 $, a known constant, where the ...
yourlazyphysicist's user avatar
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2 answers
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How to calculate the derivative of a Newtonian potential inside a box with uniform source distribution?

I'm working on a potential flow problem. I have a box, centered at the origin (i.e. $V=[-x_b,x_b]\times[-y_b,y_b]\times[-z_b,z_b]$) that has inside of it a uniform distribution of source strength. We ...
byl's user avatar
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Chetaev theorem for discrete time

In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations Theorem 1 seems to prove a discrete-time analog of Chetaev ...
xyz'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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Indeterminate Form for Partial Derivative of Flow Variables

The following is a paraphrased version of the derivation within John D. Anderson's Fundamentals of Aerodynamics's section on the Method of Characteristics: The exact governing equation for two-...
Jacob Ivanov'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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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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Riesz potential cannot be extended to $L^1 \rightarrow L^{\frac{n}{n-s}}$ bounded operator

Riesz potential is defined as $$I_s(f) = K_s*f$$ for Schwartz function $f$ with $K_s(x) = c_s|x|^{-n+s}$. By weak Young's inequality, $I_s$ can be extended to $L^p \rightarrow L^q$ bounded operator ...
Luke's user avatar
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How to verify ihe Interpolation inequality for the weighted Bessel potential spaces?

I am trying to prove the following: Let $w$ be an admissible weight, $p_1,p_2\in[1,\infty)$, $\alpha_1,\alpha_2\in\mathbb{R}$, $\theta\in(0,1)$ and \begin{equation} \alpha=\theta\,\alpha_1+(1-\theta)\...
Azam's user avatar
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Different definitions of conformal capacity

A condenser is a pair $(D,E)$, where $D$ is a domain in the plane and $E$ is a compact subset of $D$. The capacity of the condenser $(D,E)$ is defined by: $$\text{cap}(D,E) = \inf \int_{D} |\nabla u|^...
porridgemathematics's user avatar
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How to evaluate a volume integral with a singular interior point?

I would like to evaluate the following integral: $I = \int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} \frac{x-x_0}{[(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2]^{3/2}} dz dy dx$ where the fixed point $(x_0,...
byl's user avatar
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1 answer
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Level set of green function

$\textbf{Background for problem statement}$: Let $B \subset \mathbb{C}$ be a bounded domain, and $g_{B}(z,z_0) = g$ its Green's function with pole at $z_0 \in B$, so $g$ is harmonic in $B \setminus \{...
porridgemathematics's user avatar
4 votes
1 answer
136 views

Integral for the surface area of a half n-sphere

I am trying to evaluate the following integral on $\mathbb{R}^{n-1}$ $$\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n}{2}}}dx$$ I claim that this is equal to the half the surface area of the ...
Geekernatir's user avatar
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Green function for the upper-half space $\mathbb{R}^{n+1}_+$

Given $\mathbb{R}^{n+1}_+=\{X=(x,t): x \in \mathbb{R}^n , t>0 \}$ as domain, what is the explicit formula for the Green function $G(X,Y)$ for the Laplacian on $\mathbb{R}^{n+1}_+$? I know that ...
XIII's user avatar
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1 answer
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proof related to jump relation in double-layer potential

Assume $D \subset \mathbb{R}^{n}$ be a bounded and open domain with $C^{2}$ boundary. Let $x \in \partial D$ and $r>0$. Define \begin{align*} C_{r} := D \cap \partial B( x,r) \text{ and } \...
000000000's user avatar
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Can we construct a compact complete pluripolar set?

We know that analytic sets are not relatively compact in $\mathbb{C}^n$ when $n\geq 2$. In fact, we can construct plurisubharmonic functions by holomorphic functions. So analytic subset belong to ...
YaoYao Hu's user avatar
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56 views

Prove $ \lim_{\varepsilon\to 0} \frac{dH^{n-1}(\Omega\cap\partial B(x,\varepsilon))}{dH^{n-1}(\partial B(x,\varepsilon))} = \frac{1}{2}.$

Assume $\Omega\subset \mathbb{R}^n$ is a bounded open set with Lipschitz boundary; let $d H^{n-1}$ be the Hausdorff measure on $\partial \Omega$. Since $\Omega$ has Lipschitz boundary, for $d H^{n-1}$-...
Milk's user avatar
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2 votes
1 answer
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Divergence free field from neural network with automatic derivatives and a potential function

Objective My objective is to use a neural network to create some tensor field that is automatically divergence free in 2D. Background Math Many parts of physics require some kind of field to be ...
Harmsen's user avatar
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2 answers
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Evaluating this integral of the Coulomb interaction

I have the following definite integral over 6D space, in spherical coordinates: $$\int d^3r d^3r' \frac{e^{-2a(r+r')}}{|\mathbf{r-r'}|}$$ I am unsure how to approch this but I have a couple of ideas. ...
cyfirx's user avatar
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Classical balayage

Suppose $G \subset \mathbb{C}$ is a bounded domain, such that $\partial G$ has positive capacity, and let $v$ be a unit mass measure compactly supported in $G$. Then the balayage problem is to find a ...
porridgemathematics's user avatar
9 votes
0 answers
211 views

Probability, potential theory and complex analysis

The connection between Markov processes and Potential Theory is well known, as is conformal invariance of Brownian motion which allows probabilistic proofs of statements in Complex Analysis, like ...
ReLonzo's user avatar
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Perron's method for the Dirichlet problem: Necessity of subharmonic barrier

I am not seeing how to adapt my argument to produce a proof for the following statement: Let $D \subset \mathbb{C}_{\infty}$ omit at least two points, and suppose that $\infty \notin \partial D$ and $\...
porridgemathematics's user avatar
1 vote
1 answer
67 views

Potential and integral (electrodynamics)

I'm supposed to find the vector potential $A(x)$ the field generated by an infinite conducting wire of section $\pi a^2$, in which flows a constant current density $j=j \cdot \theta(a-r)\hat{z}$ ...
Tomy's user avatar
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2 votes
1 answer
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Is there a solution to the Dirichlet problem for elliptic differential equations, based on green functions?

Assume that $L$ is an elliptic operator of the form $$L(u) = \sum_{i,j} a_{ij} \partial_i \partial_j u + \sum_i b_i \partial_i u$$ where $a_{i,j}, b_i$ are measurable functions on a bounded domain $\...
MikeTeX's user avatar
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1 vote
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Is the existence problem with Dirichlet/Neumann boundary conditions for ellipitic equations solved?

Assume that $L$ is an elliptic operator of the form $$L(u) = \sum_{i,j} a_{ij} \partial_i \partial_j u + \sum_i b_i \partial_i u$$ where $a_{i,j}, b_i$ are continuous functions on a bounded domain $\...
MikeTeX's user avatar
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2 votes
1 answer
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How would you simplify this expression involving complex numbers and a logarithm?

The expression is $$\log\left(\left| \frac{z+ia}{z-ia}\right|\right) $$ Here, $a>0$ is real, $i$ is the imaginary unit, and $z=x+iy$. The expression comes from extracting the imaginary part of an ...
Chillpadde's user avatar
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1 answer
182 views

Fourier series for a potential with boundary conditions on cylinder

I'm working with a problem from Electromagnetics where I'm supposed to calculate the potential distribution in two different intervals. However, I'm very unfamiliar with the boundary conditions and I ...
Tanamas's user avatar
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3 votes
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Applications of Pluripotential Theory in real world

I am reading for a math PhD with research in Pluripotential Theory (a subfield in Several Complex Variables). I particularly do study and develop theory related to extremal functions associated with a ...
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3 votes
1 answer
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Green's First and Second Identities

I was doing some research on potential theory and Green's identities and noticed that most of the literature I find on the subject tends to define the vector field $$\tag{1}\vec F=\phi\,\text{grad}(\...
whitenoise's user avatar
1 vote
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100 views

Invariance of the Poisson integral under inversions

Let us denote the Poisson kernel on $B_r$ by $$ P(x,\zeta)=\frac{r^2-\vert x \vert^2}{r\omega_{n-1}\vert x-\zeta \vert^n}$$ where $x\in B_r$ and $\zeta\in\partial B_r$. Given a boundary function $f$ ...
Dilemian's user avatar
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1 vote
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Fractional integral related to stable process on half-space

I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
Sonny Medina's user avatar
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95 views

Solving poisson equation in polar coordinates for a separable potential.

I would like to get a step-by-step solution of the Poisson equation (in polar coordinates) below $$\nabla^2\psi(r, \phi) = 2 k(r, \phi)$$ where $\psi(r, \phi)$ is seperable (i.e. $\psi(r, \phi) = f(r)...
Sketos's user avatar
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Electrostatic potential of a charge in a 3-torus

Consider the 3-torus which arises from taking $R^3$ and identifying two points $x\equiv x+nL$ whenever $n$ is a vector with integer components. I'm curious about finding the electrostatic potential ...
Matt Dickau's user avatar
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Constructing superharmonic functions

Suppose $U$ is an open set in $\mathbb{R}^n$ ($n>1$), and $u: U\to (-\infty,\infty]$ locally bounded below. Suppose $u$ satisfies the superharmonic mean value inequality, i.e. for all ball $B(x,r)$...
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Non-tangential limits and contour integration

Before I state my actual question, I think it is important to clearly outline the definitions I am working with. The below definitions can be found in the book Logarithmic Potentials with External ...
porridgemathematics's user avatar
1 vote
0 answers
40 views

When does a plurisubharmonic function belongs to Sobolev space?

Let u be a plurisubharmonic function defined on the unit ball $\mathbb{B}$ of $\mathbb{C}^{k}$ such that $u \ge 1$. Question : why the partial derivates $\frac{\partial u}{\partial x_{i}}$ (which are ...
Analyse300's user avatar
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0 answers
55 views

Convolution of harmonic function $h$ constant implies that $h$ is constant?

Let $C\subseteq \mathbb{R}^d$ be a compact set and $\rho\in L^1(\mathbb{R}^d,[0,+\infty))$ a function whose essential support is $C$. For some $R\in (0,+\infty)$, let $U:=C+B(0,R)$ and $h:U\to \mathbb{...
5th decile's user avatar
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Conceptual Question

Hi guys I am a engineering student(in my 4th semester). I was having trouble with the following question. Kindly tell me how to approach and solve this question. Also please tell me some nice online ...
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