# 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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### 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 ...
1 vote
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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 ...
1 vote
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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)\...
1 vote
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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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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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 $$w(z)=\frac{Q}{2\... 1 vote 0 answers 73 views ### On a Riemannian manifold which is globally diffeomorphic to \Bbb R^n, can we write the Laplace-Beltrami in the prescribed form on the whole space? I have this one confusion. Let G be a Riemannian manifold which is globally diffeomorphic to \Bbb R^n . Then can we write its Laplace-Beltrami L in the 'usual elliptic form' i.e.$$\sum_{i,j=1}...
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. ...
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(...