# Questions tagged [potential-theory]

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### Extension of the Kelvin transform

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
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### Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator. Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
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### Superharmonic extension 2

This question is a simplified version of the one in the MO post Superharmonic extension. Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
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### Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice

This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $\Bbb{YF}$, namely: Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality ...
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### Inverse of the Riesz potential of a measure

Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$, $$I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy.$$ Assuming $f$ ...
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### Subharmonic functions with reasonably non absolute continuous laplacian

Does it exist a compact supported measure $\mu$ in the plane $\Bbb R^2$ with the following two properties? 1) Points have $\mu$-measure zero, 2) If $\Delta u=\mu$, then the polar set of $u$ has ...
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### A question on the problem of Dirichlet 2

Let $U$ be an open set in $\mathbb{R}^{n}$ with $n\geq2$ and $V$ an open set containing the boundary $\partial U$ of $U$. Suppose $u$ is subharmonic on $V$. We know that the generalized solution of ...
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### A question on the problem of Dirichlet

Suppose $U$ is an open set in $\mathbb{R}^{n}$ ($n\geq2$) whose complementary is not polar, and $f$ is a real-valued function defined at least on the boundary of $U$. We know that the generalized ...
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### Limit for series of Bessel functions evaluated at zeros

The following series arises in an electrostatics problem for a conducting cylinder: $$V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}$$ where $J_i$ is the Bessel function of $i^{th}$ ...
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### Sphere inversion in Riesz potential

I am reading the paper: ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554). On page 546, the authors ...
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### Extension of superharmonic functions

Let $V$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $W$ be an open neighborhood of the boundary $\partial V$ of $V$. If $u$ is superharmonic on $W$, is there a way to extend $u$ to a ...
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### A set of zero harmonic measure 2

Let 𝑉 be a bounded open set in $\mathbb{R}^{m}$, $m\geq 2$, and 𝑊 the interior of the closure of 𝑉. Let 𝐸 be a subset of ∂𝑉∩𝑊 (∂ means boundary) such that: 1) $E$ has positive ($m-$dimensional) ...
Lemma. Let $\mu$ be a measure in $\mathcal{M}(\Omega)$ and let $(v_{n})$ be a sequence of functions in $W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ converging to a function $v$ in the weak topology of ...