# 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 ...
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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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### 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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### 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 ...
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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 ...
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### 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 ...
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### 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{...