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.
347
questions
0
votes
0
answers
10
views
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 ...
- 409
1
vote
1
answer
42
views
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 ...
- 731
1
vote
0
answers
31
views
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)\...
- 11
1
vote
0
answers
145
views
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:
$$\text{cap}(D,E) = \inf \int_{D} |\nabla u|^...
- 2,060
0
votes
1
answer
47
views
How to evaluate a volume integral with a singular interior point?
I would like to evaluate the following integral:
$I = \int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} \frac{x-x_0}{[(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2]^{3/2}} dz dy dx$
where the fixed point $(x_0,...
- 33
0
votes
1
answer
43
views
Level set of green function
$\textbf{Background for problem statement}$:
Let $B \subset \mathbb{C}$ be a bounded domain, and $g_{B}(z,z_0) = g$ its Green's function with pole at $z_0 \in B$, so $g$ is harmonic in $B \setminus \{...
- 2,060
4
votes
1
answer
85
views
Integral for the surface area of a half n-sphere
I am trying to evaluate the following integral on $\mathbb{R}^{n-1}$
$$\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n}{2}}}dx$$
I claim that this is equal to the half the surface area of the ...
- 45
0
votes
1
answer
64
views
Green function for the upper-half space $\mathbb{R}^{n+1}_+$
Given $\mathbb{R}^{n+1}_+=\{X=(x,t): x \in \mathbb{R}^n , t>0 \}$ as domain, what is the explicit formula for the Green function $G(X,Y)$ for the Laplacian on $\mathbb{R}^{n+1}_+$? I know that ...
- 393
0
votes
1
answer
51
views
proof related to jump relation in double-layer potential
Assume $D \subset \mathbb{R}^{n}$ be a bounded and open domain with $C^{2}$ boundary. Let $x \in \partial D$ and $r>0$. Define
\begin{align*}
C_{r} := D \cap \partial B( x,r) \text{ and } \...
- 497
1
vote
0
answers
49
views
Can we construct a compact complete pluripolar set?
We know that analytic sets are not relatively compact in $\mathbb{C}^n$ when $n\geq 2$. In fact, we can construct plurisubharmonic functions by holomorphic functions. So analytic subset belong to ...
- 11
1
vote
0
answers
50
views
Prove $ \lim_{\varepsilon\to 0} \frac{dH^{n-1}(\Omega\cap\partial B(x,\varepsilon))}{dH^{n-1}(\partial B(x,\varepsilon))} = \frac{1}{2}.$
Assume $\Omega\subset \mathbb{R}^n$ is a bounded open set with Lipschitz boundary; let $d H^{n-1}$ be the Hausdorff measure on $\partial \Omega$.
Since $\Omega$ has Lipschitz boundary, for $d H^{n-1}$-...
- 625
2
votes
1
answer
47
views
Divergence free field from neural network with automatic derivatives and a potential function
Objective
My objective is to use a neural network to create some tensor field that is automatically divergence free in 2D.
Background Math
Many parts of physics require some kind of field to be ...
- 21
1
vote
2
answers
151
views
Evaluating this integral of the Coulomb interaction
I have the following definite integral over 6D space, in spherical coordinates:
$$\int d^3r d^3r' \frac{e^{-2a(r+r')}}{|\mathbf{r-r'}|}$$
I am unsure how to approch this but I have a couple of ideas.
...
- 35
0
votes
0
answers
16
views
Velocity potential in conical coordinate
Velocity potential in conical coordinate
I just want to make sure that the equation regarding velocity potential in conical coordinates (the above picture) is correct or if there is a typo. I came ...
2
votes
0
answers
41
views
Classical balayage
Suppose $G \subset \mathbb{C}$ is a bounded domain, such that $\partial G$ has positive capacity, and let $v$ be a unit mass measure compactly supported in $G$. Then the balayage problem is to find a ...
- 2,060
9
votes
0
answers
169
views
Probability, potential theory and complex analysis
The connection between Markov processes and Potential Theory is well known, as is conformal invariance of Brownian motion which allows probabilistic proofs of statements in Complex Analysis, like ...
- 187
0
votes
1
answer
101
views
Perron's method for the Dirichlet problem: Necessity of subharmonic barrier
I am not seeing how to adapt my argument to produce a proof for the following statement:
Let $D \subset \mathbb{C}_{\infty}$ omit at least two points, and suppose that $\infty \notin \partial D$ and $\...
- 2,060
1
vote
1
answer
66
views
Potential and integral (electrodynamics)
I'm supposed to find the vector potential $A(x)$
the field generated by an infinite conducting wire of section $\pi a^2$, in which flows a constant current density $j=j \cdot \theta(a-r)\hat{z}$ ...
- 429
0
votes
0
answers
29
views
Riesz potential with variable exponent
Does the following hold for Riesz potentials in $\mathbb{R}^n$?
There exist a function $\beta:\mathbb{R}^n \to \mathbb{R}$ such that $\alpha(x)>0$ and
$$ I_\alpha f(x)=\int_{\mathbb{R}^n} \frac{f(y)...
- 13
2
votes
1
answer
199
views
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 $\...
- 1,774
1
vote
0
answers
45
views
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 $\...
- 1,774
2
votes
1
answer
65
views
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 ...
- 45
0
votes
0
answers
37
views
Complex potential of a linearly varying vortex panel
As shown in the image below, Zj, Zj+1, Zj-1, D,E, and Z are points in the complex plane.
There is a vortex with a strength $\gamma$ (I in the picture) placed on Zj such that the vortex strength is ...
0
votes
1
answer
142
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 ...
- 1,827
2
votes
0
answers
80
views
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 ...
- 5,715
3
votes
1
answer
521
views
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
$$\tag{1}\vec F=\phi\,\text{grad}(\...
- 387
1
vote
0
answers
66
views
Invariance of the Poisson integral under inversions
Let us denote the Poisson kernel on $B_r$ by $$ P(x,\zeta)=\frac{r^2-\vert x \vert^2}{r\omega_{n-1}\vert x-\zeta \vert^n}$$ where $x\in B_r$ and $\zeta\in\partial B_r$. Given a boundary function $f$ ...
- 1,225
1
vote
0
answers
18
views
Fractional integral related to stable process on half-space
I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
- 11
0
votes
0
answers
73
views
Solving poisson equation in polar coordinates for a separable potential.
I would like to get a step-by-step solution of the Poisson equation (in polar coordinates) below
$$\nabla^2\psi(r, \phi) = 2 k(r, \phi)$$
where $\psi(r, \phi)$ is seperable (i.e. $\psi(r, \phi) = f(r)...
- 53
1
vote
0
answers
30
views
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 ...
- 2,157
0
votes
0
answers
37
views
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)$...
- 1,369
1
vote
0
answers
65
views
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 ...
- 2,060
1
vote
0
answers
34
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 ...
- 67
1
vote
0
answers
42
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{...
- 2,175
0
votes
0
answers
47
views
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 ...
1
vote
1
answer
100
views
Coulomb Potential and laplace operator
How to proof the equality $\int_{S_{a}^{n-1}}\nabla u.d\sigma= c *vol (S^{n-1})$.
Hi all,
I am reading a text in portugues about PDE, is about Laplace operator and Coloumb Potential my specifics ...
- 1,219
1
vote
0
answers
31
views
Volume potential for Laplace equation in $\mathbb{R}^3$ with $L^p$ source term
It is known that the solution of
$$-\Delta u = f \quad \text{in } \mathbb{R}^3$$
can be represented through the volume potential
$$ Vf(x) = \int_{\mathbb{R}^3} E(x-y)f(y) \, dy, \quad x \in \mathbb{R}^...
- 2,488
0
votes
1
answer
313
views
Equilibrium measure on a real segment
This is probably an elementary question but I still cannot convince myself or find a decent argument to believe.
Could anyone please explain to me why do we believe that the arcsine distribution (or ...
- 257
1
vote
0
answers
25
views
Existence of descending path in landscape (without 'prominences')
Let $\mu$ be a Borel measure on $\mathbb{R}^d$ ($d>2$) so that
$$\int_{\mathbb{R}^d} \frac{d\mu(x)}{1+|x|^{d-2}}<+\infty$$
The potential
$$\Phi(x):=\int_{\mathbb{R}^d} \frac{d\mu(y)}{|x-y|^{d-2}}...
- 2,175
2
votes
2
answers
135
views
Second-order nonlinear ODE $d^{2}y/dx^{2} =y^{-1/2}f(x)$
Has this second-order nonlinear ODE been studied before or been given a known name that I can look up for further study or investigation?
$$ y’’ \sqrt{y}+f(x)=0, $$
in single real variable $x$, with $...
- 1,299
0
votes
1
answer
135
views
To show that a complex function when restricted to one variable is convex, implies it is subharmonic.
This is a problem from T. Ransford's Potential Theory in the Complex Plane.
Let $u:\Delta(0,r)\to\mathbb{R}$ be a function such that $u(x+iy)$ is convex in $x$ for each fixed $y$ and convex in $y$ for ...
- 325
2
votes
0
answers
59
views
Average behavior of Green's function near the boundary
Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for ...
- 317
0
votes
1
answer
79
views
$u\in C_0^3(\Omega)$ satisfis $ -\Delta u=f(x), x\in\Omega. $Prove that $ \sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} dx\leq n\int_{\Omega}f^2 dx. $
Suppose $u\in C_0^3(\Omega)$ satisfis
$$
-\Delta u=f(x), x\in\Omega.
$$
Prove that
$$
\sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} \mathrm{d}x\leq n\int_{\Omega}f^2\mathrm{d}x.
$$
I have tried as ...
- 1,335
3
votes
0
answers
309
views
Potential flow around ellipse
I stuck at the following problem:
Consider a planar incompressible potential flow (i.e., an irrotational flow) around an
elliptic profile
\begin{equation*}
P = \lbrace (x,y) \in \mathbb{R}^2 | \frac{...
- 1,040
1
vote
1
answer
118
views
Sum of subharmonic functions attaining a maximum in an open, connected set.
I want to prove the following statement: If $p_1,..,p_n$ are subharmonic functions on an open, connected subset $V$ of the complex plane and if $p_1+...+p_n$ attains a maximum on $V$, then each $p_i$ ...
user879426
0
votes
0
answers
22
views
How to show that $\int_{B(x, t)} \Delta f d \lambda=t^{N-1} \int_{S} \frac{\partial}{\partial t} f(x+t y) d \sigma(y)$?
Let $f$ be a $C^{2}$ function on an open set which contains $\overline{B(x, r)} .$ We can use first Green's formula and then differentiation under the integral sign to obtain
$$
\int_{B(x, t)} \Delta ...
- 31
2
votes
2
answers
39
views
A series sum in a potential flow problem
A point source is located at the origin in a complex plane and two parallel walls are held at $\pm ib$. The complex potential should be solved by the method of image. The sum takes
$$
w(z)=\frac{Q}{2\...
- 31
1
vote
0
answers
73
views
On a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$, can we write the Laplace-Beltrami in the prescribed form on the whole space?
I have this one confusion.
Let $G$ be a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$ . Then can we write its Laplace-Beltrami $L$ in the 'usual elliptic form' i.e. $$\sum_{i,j=1}...
- 2,254
2
votes
1
answer
335
views
Harmonic function extension and Conformal maps
Let $\bigcup_{j=1}^n [0,a_i] = S$ be a 'star' where $a_i \in \mathbb{C}$ and the $[0,a_i]$ denote the line segments from $0$ to $a_i$ in the plane, all of the $a_i$ here are distinct and nonzero. ...
- 2,060
0
votes
1
answer
78
views
Positive function which is harmonic in every compact set in a cone
Let $C \subset \mathbb R^3$ be an open unbounded cone with vertex at the origin (I do not care much about the angle of the cone but assume that $C \cap \partial B(0,1) = B((0,0,1), r) \cap \partial B(...
- 317