# Tag Info

## Hot answers tagged potential-theory

### 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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### 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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### 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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### $\log |f|$ 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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### $\log |f|$ 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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### 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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### 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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### 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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### 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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### 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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### 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 $$(E_n * \si_{aS^{n-1}})(x)=a(E_n * \si_{S^{n-1}})(x/a) \tag{1}\label{1}$$...
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### 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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### 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 ...
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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 ...

### 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 ...
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