# Questions tagged [potential-theory]

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135
questions

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### Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential

Let $k(t,x)$ be the transition density of Brownian motion $$ k(t,x) := \frac{1}{\sqrt{2 \pi t}} \exp \left\{ \frac{-x^2}{2t} \right\} , \quad t \geq 0, x \in {\mathbb R.}$$
Question
Let $0 < x &...

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### Stability and capacity, error in the book of Adams-Hedberg?

I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...

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113 views

### Subharmonic in any holomorphic coordinates = Plurisubharmonic?

An upper semi-continuous function $u : \Omega \to \mathbb{R}$, $\Omega \subseteq \mathbb{C}^n$ is said to be subharmonic if it satisfies the submean inequality $u(a) \leq \mu_S(u;a,r)$, where $\mu_S(...

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76 views

### Conformal isomorphism uniquely determined by boundary identification?

Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...

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58 views

### Weak maximum principle for a perturbation of the Laplacian

This question is motivated by similar considerations for the Kohn-Laplacian in the Heisenberg group, but it seems that I cannot even give an answer in the Euclidean case, so here we go.
Suppose that ...

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132 views

### Extension of subharmonic function: can someone explain the details?

In this paper we have the following situation on page 60. $E$ is a compact subset of $\mathbb{R}^\tau\cup\{\infty\}$ (one point compactification) for $\tau\geq2$, $M_0$ is a point in the boundary of $...

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44 views

### The converse of a Poincaré's result on regular boundary points

Let $V$ be a bounded open set in $\mathbb{R}^n$ with $n>1$. According to a well known result due to Poincaré, if $x$ is a point in the boundary $\partial V$ and there exists a ball $B$ such that $x\...

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95 views

### Dirichlet problem for a subharmonic function

Suppose $K$ is a compact subset of $\mathbb R^n$ , $V_0$ and $V_1$ the complements of $K$ in $\mathbb R^n$ a and $\mathbb R^n_\infty$ (one point compactification), respectively. Let $u$ be ...

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113 views

### Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit sphere

This is a follow-up question to the one asked here (the unit circle case). What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^2}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The ...

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302 views

### Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle

What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The answer appears to be uniform measure, since informally it appears better to have ...

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91 views

### Extension of subharmonic functions at infinity

Let $W$ be the complement of a compact set $K$ in $\mathbb{R}^{n}$, and $u$ a subharmonic function on $W$. Can we find, under some conditions, a function $\tilde{u}$ that is subharmonic on $W\cup\{\...

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44 views

### 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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133 views

### 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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81 views

### 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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63 views

### 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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66 views

### 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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38 views

### 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) ...

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104 views

### Convergence in unbounded domains

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 ...

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96 views

### A set of zero harmonic measure

We know that a set may be of zero harmonic measure without its Lebesgue measure being zero (see Armitage and Gardiner, classical potential theory, pg 178).
Now, consider the following problem. Let $...

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161 views

### Minimizing energy on $\mathbb{S}^2$ for absolutely monotonic type potentials

For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a ...

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89 views

### Harmonic functions with boundary condition

I have a question on a harmonic function and the boundary behavior.
Let $\mathbb{U} \subset \mathbb{C}$ be a unit disk. We denote by $\overline{\mathbb{U}}$ the closure of $\mathbb{U}$ in $\mathbb{C}$...

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36 views

### Hitting order of sets by a Lévy process

Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything ...

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220 views

### Thinness and polarity

Let $D$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $E$ a subset of the boundary $\partial D$ of $D$. $D$ is said to be thin at a point $y\in D$ if there is a superharmonic function $u$...

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124 views

### Capacity and harmonic measure

Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ with $n>1$ and $E$ a subset of its boundary. We know that if $E$ has capacity zero I.e. it is a polar set , then the harmonic measure of $E$ ...

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58 views

### Sequence of harmonic measure

There is a well-known result stating that if $\mu_{n}$ is a sequence of uniformly bounded measures on a compact set $E$ of $\mathbb{R}^{m}$, then there is a subsequence $\mu_{n_{j}}$ that converges ...

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65 views

### A question about harmonic measure 2

Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E$ a subset of the boundary of $V$. If $\omega(x,W)(E)=0$ ($\omega(x,W)$ is ...

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50 views

### A question on harmonic measure

Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E\subset W$. If $\omega(x,V)(E)=0$ ($\omega(x,V)$ is the harmonic measure of $...

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164 views

### Examples of polar sets

I would like to know some examples of the following polar sets (if they exist):
a non trivial uncountable polar set in $\mathbb{R}^{2}$;
a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with ...

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73 views

### Reference for a theorem on subharmonic functions

I need a reference for a theorem that states: Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. Let $u$ be a subharmonic function on $D$. ...

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104 views

### A question about Riesz decomposition theorem

Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. ...

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201 views

### Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...

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48 views

### Extension of a $\delta$-subharmonic function that is subharmonic on a reduced domain

Suppose $B$ is a ball in $\mathbb{R}^{m}$ and $u$ and $s$ are subharmonic on $B$. Suppose there is a closed subset $F$ of the closure of $B$ with no interior such that $v=u-s$ is subharmonic on $B\...

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299 views

### multivalued holomorphic function on Riemann surfaces

Let $M$ be an open Riemann surface and $f$ a multivalued holomorphic function from $M$ to $\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. Suppose that the monodromy of $f$ lies in the two-...

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101 views

### What fraction of a charge is induced on a surface via balayage?

Consider a smooth, bounded domain $\Omega \subset \mathbb{R}^3$, and place a charge $q>0$ at a point $z\in\mathbb{R}^3\setminus\overline\Omega$. Via the concept of balayage, there is an induced ...

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### Does an integration by parts formula hold for the spectral fractional Laplacian in 1-d?

Is there an integration by parts formula for the spectral fractional Laplacian in a bounded interval $[a,b] \subset \mathbb R$?

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### Sets having large capacity

The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework.
Let ${\...

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58 views

### Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...

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61 views

### Measure for which it's logarithmic potential is continuous

Let $\mu$ be a compactly supported borel probability measure on $\mathbb C$ then it's logaruthmic potential is,
$P_{\mu}(z)= \int_{\mathbb C} log|z-w|d\mu(w)$
It's well known that $P_\mu$ is ...

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164 views

### Conformal mappings and its singularity

I have a question about singularities of conformal mappings.
Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to ...

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161 views

### Motivation for study of parabolic manifolds

Why separate the complete Riemannian manifolds that admit and those that do not admit positive Green function?
In summary, what is the motivation for studying parabolic and non-parabolic manifolds?

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131 views

### Most general conditions for (weak or classical) solutions to Poisson's equation

I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving
$$
\left\{
\begin{align}
& \Delta \psi = - \rho & & \mbox{in } \mathbb{...

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163 views

### Quasinilpotent vectors of Newton potential vanish

Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential
\begin{equation}
T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy.
\end{equation}
It is well know ...

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114 views

### Continuity of harmonic functions

I have a question about harmonic functions with respect to symmetric Markov processes.
Let $E$ be a locally compact separable metric space and $\mu$ a positive Radon measure on $E$.
Let $X=(\{X_t\}...

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206 views

### Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...

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80 views

### Riesz measure of a smooth subharmonic function on a ball

Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...

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68 views

### Exit time of $\alpha$-stable process to an open set and to its closure

Let $W$ be a symmetric $\alpha$-stable process with its generator
$-(-\Delta)^{\alpha/2}$ for some $\alpha \in (0, 2]$ under $\mathbb P$. Let $\mathbb P^x$ be the probability measure induced by a ...

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109 views

### Differentiability of the logarithmic potential

Assume $\mu$ is a measure supported on a real finite interval $[a,b]$, and let
$$p_\mu(z)=\int\log|z-t|d\mu(t),$$
denote the logarithmic potential associated to $\mu$. Are there (possibly simple) ...

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170 views

### Comparison of Bessel Capacities

The Bessel kernels $G_{\alpha},\, \alpha>0$ are defined by their Fourier transform
$
\hat G_{\alpha}(\xi):= \frac 1 { (1+4\pi ^{2}\vert \xi \vert ^{2})^{\alpha/2}}.
$
Bessel $(\alpha, p)$-capacity ...

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211 views

### Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...

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111 views

### Boundedness of a finite subharmonic function

Let $$u\colon B^n(0,1)\to \mathbb{R}$$ be a subharmonic function in the open unit ball in $\mathbb{R}^n$. The crucial assumption is that $u$ never equals $-\infty$.
Is it true that $|u|$ is ...