Tagged Questions

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0answers
61 views

Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the …
4
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0answers
95 views

A finely open set, not open up to polar set?

I already asked this on M.SE, but get no answers. Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge …
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1answer
71 views

Higher dimensional analogue of Kellog’s theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the …
1
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1answer
383 views

Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \ …
10
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1answer
274 views

Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabrie …
3
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1answer
246 views

Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, whic …
4
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1answer
275 views

Logarithmic potential

Given a continuous, compactly supported function $f$ on $R^2$, it is known that the logarithmic potential of $f$, that is $$ U_{f}(x):=-\frac{1}{2\pi}\int\log|x-y|f(y)dy $$ has the …
0
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1answer
124 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. T …
5
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2answers
441 views

Kähler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone …
1
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1answer
199 views

Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\part …
12
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2answers
887 views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course i …
4
votes
2answers
350 views

Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times {0}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator. We wish to solve the D …
4
votes
0answers
292 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that t …
0
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1answer
145 views

Properties of the quadruple layer potential

What is known about the quadruple layer potential in 3D (on closed smooth surfaces)? In terms of jump relations, continuity on Hölder Spaces (and/or Sobolev spaces), and Calderon-t …
3
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1answer
332 views

Green’s formula for a Markov process

For a Markov process $X$ on the Polish space $\mathscr X$ its transition probability is given by $$ P(x,A) :=\mathsf P_x (X_1\in A) $$ and $X$ is time-reversible if there is a prob …

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