Questions tagged [potential-theory]

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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 &...
5
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0answers
52 views

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 ...
2
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1answer
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(...
2
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1answer
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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0answers
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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0answers
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 $...
2
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1answer
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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1answer
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 ...
2
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0answers
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 ...
5
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1answer
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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1answer
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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1answer
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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1answer
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 ...
4
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1answer
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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1answer
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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1answer
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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0answers
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 ...
2
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1answer
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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1answer
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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0answers
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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2answers
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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1answer
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 $E$ ...
2
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0answers
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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1answer
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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0answers
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 $...
2
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2answers
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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1answer
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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1answer
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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1answer
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 ...
2
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0answers
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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1answer
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-...
3
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1answer
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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0answers
77 views

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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0answers
115 views

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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0answers
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 ...
3
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1answer
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 ...
2
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1answer
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 ...
3
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1answer
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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2answers
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{...
3
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1answer
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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1answer
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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1answer
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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1answer
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 $\mu$...
1
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1answer
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 ...
2
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1answer
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) ...
1
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1answer
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 ...
5
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0answers
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{...
1
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1answer
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 ...