# Tag Info

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 ...
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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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### 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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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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### $\log |f|$ 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 ...
• 77.2k
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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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### 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 ...
• 52k
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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 ...
• 77.2k
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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 ...
• 22.5k
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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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### 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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### 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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### 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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### 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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### 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 ...
• 77.2k