10
votes
Books about capacity theory
I think the best treatment of basic facts about capacity from the perspective of Sobolev spaces is in Chapter 4 of
L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised ...
- 26.4k
9
votes
Accepted
Calculation of logarithmic capacity?
In two dimensions, you have a powerful tool, the Riemann mapping. If you have a compact set in the plane whose complement is connected, knowing explicitly
the map of the complement onto the exterior ...
- 87.2k
9
votes
Canonical English edition of Dellacherie and Meyer's "Probabilities and Potential"
The original French edition of Probabilities and Potential contains 24 chapters spread over five volumes, as outlined here. The English translation of chapters 1 through 13 is spread over three ...
- 172k
9
votes
Accepted
$\log |f|$ is subharmonic
Conditions for a log-subharmonic function $f$ on $D\in\mathbb{R}^n$ are described by Mochizuki in A Class of Subharmonic Functions and Integral Inequalities (2004).
The conditions are phrased in terms ...
- 172k
9
votes
$\log |f|$ is subharmonic
A real-valued function $u\geq 0$ such that $\log u$ is subharmonic is called logarithmically subharmonic. One easy theorem is that the sum of logarithmically subharmonic functions is also ...
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8
votes
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A problem of potential theory arising in biology
The answer is negative.
Assume that $K_{0}$ and $K_{1}$ are two disjoint balls. If $K_{0}$ is at least twice as big as $K_{1}$, then $K_{0}$ should keep some nonzero distance in order to minimize ...
- 3,681
8
votes
Accepted
Capacity of a unit disk with a small bump
Both questions have exact explicit answer, which is explained in any good textbook on analytic functions. A region on the Riemann sphere is called a digon if its boundary consists of
two arcs of ...
- 87.2k
6
votes
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Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle
Yes, it is true for the circle (though the reason is not quite the one you suggested). We shall consider the discrete version of the problem, which is to put some odd number $n\ge 3$ of points (think ...
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5
votes
Accepted
A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential
There are two kinds of conditions:
a) the local one: distributional Laplacian of $V$ must be a signed measure (difference of two non-negative distributions). I do not think that there is a simpler ...
- 87.2k
5
votes
Core for a Sobolev space
Smooth functions $C^\infty(D)$ are always dense in $W^{1,p}(D)$ when $1\leq p<\infty$. This is a classical result of Meyers and Serrin and you can find it in any textbook on Sobolev spaces. However,...
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5
votes
Accepted
Newtonian potentials of balls and spheres
$\newcommand\si\sigma$Below is a verification of your results, with integration done in Mathematica.
Here I used the formulas
$$(E_n * \si_{aS^{n-1}})(x)=a(E_n * \si_{S^{n-1}})(x/a) \tag{1}\label{1} $$...
- 110k
4
votes
Differentiability of the logarithmic potential
I think there are necessary and sufficient conditions in the literature, but here is a simple sufficient condition : $d\mu=h\ dt$ with $h\in L^2([a,b],dt)$, because then the (distributional) ...
- 3,045
4
votes
Calculation of logarithmic capacity?
Here's a larger class of examples coming from dynamics. Let $$f(x)=ax^d+bx^{d-1}+\cdots+c\in\mathbb C[x]$$ be a polynomial of degree $d\ge2$. The filled Julia set of $f$ is the set
$$ K_f := \bigl\{ z\...
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4
votes
Accepted
Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$
The following result is well known. It is Theorem 5.1 in [1]. It is proved by the method of duality solutions sue to Stampacchia.
Theorem. Let $\mu$ be a signed Borel measure with the finite total ...
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4
votes
Most general conditions for (weak or classical) solutions to Poisson's equation
The name of the subject is Potential Theory (your integral (3) is called the
Newtonian potential). Good references are:
N. Landkof, Foundation of modern potential theory, Springer 1972,
M. Brelot, ...
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4
votes
Accepted
Subharmonic function in unbounded regions
Yes, this is called the Phragmen-Lindelof Principle: For every region on the Riemann sphere, if $h$ is subharmonic and bounded from above, and
$$\limsup_{z\to\zeta}h(z)\leq 0$$ for all $\zeta\in\...
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4
votes
A question on minimum principle
The question can be phrased equivalently as follows: for what $D$ there is no infinite Martin boundary point.
This will not be the case for most "typical" domains.
A simple example of an ...
- 16.2k
4
votes
Value of $\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}$
The series has no expression in terms of elementary functions, but it does represent a special function (either the incomplete beta function $B$ or the Lerch transcendent $\Phi$):
$$F(b,z)=\sum_{n=1}^\...
- 172k
4
votes
Accepted
Green's function in terms of logarithmic potential and energy of a measure
What you stated cannot be true: Green function depends only on $K$, but your $\Phi_\mu$ is the potential of an arbitrary
measure on $K$. These formulas become true when $\mu$ is the
EQUILIBRIUM ...
- 87.2k
3
votes
Accepted
A finely open set, not open up to polar set?
I like the following abstract construction, which relies on a connection between quasi-continuity and the fine topology and also works in the setting of $p\ne 2$. It is debatable whether the resulting ...
- 482
3
votes
Books about capacity theory
Ransford's textbook on this topic is great. Here is how he describes the goal of his book in the preface, which I think he accomplishes very well.
When first learning potential theory, as a new ...
- 2,115
3
votes
Most general conditions for (weak or classical) solutions to Poisson's equation
1st question : indeed, Hôlder continuity seems to be the weakest possible condition for (1), (2), (3) to hold. Gilbarg and Trudinger give an example of a continuous $\rho$ in the unit ball such that $\...
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3
votes
Motivation for study of parabolic manifolds
One that comes to mind is that parabolicity of a Riemannian manifold is equivalent to recurrence of its Brownian motion, e.g. of the Markov process whose infinitesimal generator is $\frac12\Delta$.
...
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3
votes
Accepted
Conformal mappings and its singularity
I did not know at all... I am very sorry but can you tell me any references
Ah-oh. It is hard to find a decent reference now because nobody cares about writing down such trivialities any more: they ...
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3
votes
Accepted
Measure for which it's logarithmic potential is continuous
A necessary and sufficient condition was given in
M.G. Arsove, Continuous potentials and linear mass distributions, SIAM
Rev. 2, 1960, 177-184.
Let $\mu_{r}(z)$ be the total mass of the ...
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3
votes
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Sphere inversion in Riesz potential
Well, you may like to have a look at the original M. Riesz's 1938 paper: it is a fantastic read!
In the language of these papers, a "potential" of exponent $\alpha$ is a function $f$ of the form
$$ f(...
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3
votes
Accepted
Subharmonic in any holomorphic coordinates = Plurisubharmonic?
It is actually enough to assume that your function $u$ defined on a neighborhood of $0\in \mathbb C^n$ remains subharmonic after composing with any linear transformation.
Indeed, let $0\neq \...
- 2,617
3
votes
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A possible characterization of subharmonic functions
By "$u$ is subharmonic" do you mean it is so in the comparison sense, namely: given every closed ball $B\subseteq \Omega$, and every harmonic $\phi$ on $B$ with $\phi|_{\partial B} \geq u|_{\...
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2
votes
Books about capacity theory
Also a very good book:
Title: Condenser Capacities and Symmetrization in Geometric Function Theory
Author(s): Vladimir N. Dubinin (auth.)
Publisher: Birkhäuser Basel
Year: 2014
ISBN: 978-3-0348-0842-2,...
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2
votes
Accepted
Is this a superharmonic function?
First you have to assume that $F$ has measure zero, since a superharmonic function is locally integrable.
Under this assumption, setting $u=\infty$ on $F$ results in a superharmonic function. To see ...
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