Questions tagged [potential-theory]

Question 1

Though this is a problem from physics, I think it is more about mathematics. Suppose an infinite cylinder $V$ of radius $a$ with its center aligned in $z$ axis carrying a current density $$ \...

Question 2

The fine topology of the potential theory is the coarsest topology that makes superharmonic functions continuous in the extended sense. We know that $\mathbb{R}^n$ endowed with the fine topology is ...

Question 3

I have a to find the wave-function of the following problem. I have free spineless fermions in a 2D box trap of size $L_x\times L_y$. To such problem, the solution is known and is $$\Psi(x,y) =\sqrt{\...

Question 4

The electrostatic potential $\phi\left(\boldsymbol{r}\right)$ is given as $$ \phi\left(\boldsymbol{r}\right) = \frac{1}{4\pi\varepsilon_0}\int\limits_{V'}\frac{\varrho\left(\boldsymbol{r'}\right)}{\...

Question 5

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded strictly convex domain with $C^{2}$ boundary(or smoother). Define $$u(x)=\int_{\partial \Omega}\frac{1}{|x-y|}d\sigma_{y}, \quad x\in \Omega.$$ Question: ...

Question 6

Let $D\subset\mathbb{R}^3$ be a bounded $C^2$ domain. On its boundary $\partial D$, place two disjoint, tiny patches $S_\varepsilon$ and $T_\delta$ with diameters $\varepsilon,\delta\ll1$. Consider ...

Question 7

Let $d\ge 2$. On $\mathbb{Z}^d$ define the discrete Laplacian $$ \Delta u(x)=\sum_{i=1}^d\big(u(x+e_i)+u(x-e_i)-2u(x)\big), $$ and for $S\subset\mathbb{Z}^d$ define the upper Banach density $$ d^*(S)=\...

Question 8

Let $N > 2$ and let $\omega, \Omega \subset \mathbb{R}^N$ be open and bounded sets with smooth boundary. Assume both sets contain the origin. For $\sigma > 0$, consider the boundary value ...

Question 9

I want to understand the following integral $$\int_{0}^{\sqrt{2}}\frac{\log|r-t|}{(4-t^2)^2}dt$$ for values of $r\in [0, \sqrt{2}]$. What I have tried so far is to used Laurent series expansions, but ...

Question 10

Let $M$ be a closed manifold of dimension $m$. Let $p\in C^{\infty}(M\times M\times (0,\infty); \mathbb R)$ be the heat kernel on $M$. On page 193 of Chapter 7, author defines an integral operator ...

Question 11

I am reading through the 'Geometry and Spectra of Compact Riemann Surfaces', Chapter 7. I found the following integral estimate, which seemed interesting to me and it took me a while to figure out. ...

Question 12

Real functions Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a $\mathcal{C}^\infty$-function, with Hessian matrix $H_f := \left( \frac{\partial^2 f}{\partial x_i \partial x_j} \right)_{i,j = 1}^n$. ...

Question 13

For $n\geq2$, $P\subset\mathbb R^n$ is said to be polar if there is some superharmonic function $u$ which is identically $\infty$ on $P$. These sets are generally thought of as small, but I lack ...

Question 14

I have a question about some sort of uniqueness in the context of (planar) single-layer potential theory: Suppose, $ \Omega = \Omega_{\text{in}} \subset \mathbb{R}^2$ is some bounded open subset with (...

Question 15

Currently I am reading the Book "Potential Theory in the Complex Plane" by Ransford and I try to solve Exercise 1.3.1. Before I state the exercise, note that $\Delta(0,\rho)$ denotes the ...

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Questions tagged [potential-theory]

Magnetic Field of Cylinder with Uniform Current Density

Is there a polar finely open set?

Series of potentials in box trap wave-function

Electrostatic potential formula

Critical points of the single-layer potential $u(x)=\int_{\partial \Omega}\frac{1}{|x-y|}d\sigma_{y}$ inside a strictly convex bounded domain.

Capacity–duty flux split for gated vs fixed window

Does $\Delta u\in \ell^1(\mathbb{Z}^d)$ with bounded first differences force $u$ to be affine?

Poisson's equation in a domain with a hole

Approximation of (eg, finding accurate bounds for) $\int_{0}^{\sqrt{2}}\frac{\log|r-t|}{(4-t^2)^2}dt$, for $r\in [0, \sqrt{2}]$

Understanding the integral operator $G_\sigma$ defined in Buser's book Geometry and Spectra of Compact Riemann surfaces, Chapter 7

An integral estimation form Buser's book: Geometry and Spectra of Compact Riemann Surfaces, Chapter 7.

Function with semidefinite Hessian everywhere is convex or concave

Are polar sets lower-dimensional?

Uniqueness of Single-Layer Potentials and Connection to Dirichlet-to-Neumann Operators

Application of Harnack's inequality

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