9 votes
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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 ...
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8 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 ...
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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 ...
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8 votes
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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 ...
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7 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 ...
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7 votes
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$\log |f|$ 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 ...
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7 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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6 votes
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electron configuration on manifolds

There seems to be a considerable body of work on this: Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds D.P. Hardin, , E.B. Saff1, More by Hardin and Saff. http:/...
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  • 93.7k
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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  • 52k
5 votes
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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 ...
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4 votes
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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
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Which domain maximizes the energy of the Lebesgue measure?

That a circular disk maximizes the energy for the Lebesgue measure follows immediately from Riesz's inequality: $I(f,g,h)\le I(f^*,g^*,h^*)$,where $I(f,g,h) = \langle f, g*h\rangle$, and $f^*$ is the ...
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  • 5,758
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) ...
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  • 3,025
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
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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 ...
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3 votes

Convergence in energy of bounded (semi)subharmonic functions

I'm reporting an example, presented to me by Bozhidar Velichkov, showing that the answer to my question is no. The example is based on a construction by Cioranescu-Murat given in their paper "Un ...
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3 votes
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Oscillation of subharmonic functions of slow growth

Yes, of course. Take any convergent series with positive terms $a_k$. Consider the function $u(z)=\sum a_k\log|1-z/z_k|$. This is a subharmonic function, $u(z_k)=-\infty$, and satisfies $u(z)=O(\log|z|...
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3 votes
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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 ...
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3 votes
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Is this integration by parts legitimate?

This "integration by parts" is true in larger generality: Let $\mu$ be a measure, $$u(x)=\int\frac{1}{|x-y|^{n-2}}d\mu(y).$$ Then $$\int u(x)d\mu=\int\frac{1}{|x-y|^{n-2}}d\mu(x)d\mu(y)$$ is called ...
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3 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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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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  • 3,408
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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  • 1,655
3 votes
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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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  • 52k
3 votes
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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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  • 3,408
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
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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 \...
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  • 2,522
2 votes
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uniqueness for Poisson equation in R^d with mildly regular data

If $v$ is a temperate distribution such that $\Delta v=0$ then the Fourier transform of $v$ must be supported at 0 hence it is a finite combination of derivatives of the Dirac delta, which in the end ...
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2 votes
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Positivity of logarithmic energy of certain measures

Yes, this is true if the supports of your measures are in the unit disk. All your other conditions are redundant: you do not need the curve, if you have one, it does not have to be smooth, and the ...
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2 votes
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Approximation of subharmonic functions

Yes, this is true. References: Any book on potential theory or subharmonic functions, for example, N. Landkof, Foundations of modern potential theory, W. Hayman and P. Kennedy, Subharmonic functions ...
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