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185 views

### Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...

**-1**

votes

**1**answer

110 views

### Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...

**8**

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**0**answers

113 views

### Functions between Markov chains that preserve local harmonicity

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...

**6**

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385 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 the restriction $f:...

**5**

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**0**answers

189 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 $\mathbb R^n$, $n \ge 2$, which is not ...

**4**

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**0**answers

106 views

### regularity of zero point

We consider 1-d process $X$
$$ X(t) = b t + J_{t} + M_{t}$$
where $b$ is constant, $M$ is a continuous martingale process with
$M(0) = 0$, and
$J$ is a symmestric $\alpha$-stable process with its ...

**4**

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**0**answers

945 views

### A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions.
First recall the Cauchy integral formula:
Let $U$ be an open subset of $\mathbb{...

**3**

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**0**answers

206 views

### Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...

**3**

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**0**answers

389 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 probability measure $\pi$...

**3**

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**0**answers

280 views

### Best Poincare constants on the surface of a ball

I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first ...

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122 views

### The minimum value of a energy integral

Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0
$ and
$${\nabla ^2}\...

**2**

votes

**0**answers

162 views

### Boundedness of the solution of the integral equation associated to the heat kernel

Let $\Omega$ be a bounded open set of $\mathbb{R}^n$ ($n\geq 2$), $\Omega\in C^{1+\alpha}$ with $\alpha\in(0,1)$. Let $\Phi_n$ be the fundamental solution of the heat equation, that is
$$\Phi_n(t,x)\...

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**0**answers

40 views

### Capacity approximations by sets with regular boundary

Suppose I have a continuous, compactly supported function $f : \mathbb{R}^2 \to \mathbb{R}_{+}$ and I define the set $S := f^{-1}([a,\infty)) \subset \mathbb{R}^2$ for some $a > 0$. It is a ...

**2**

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**0**answers

68 views

### Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets.
For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...

**2**

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**0**answers

183 views

### Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...

**2**

votes

**0**answers

105 views

### A Fourier analysis-type identity arising in potential theory

I have recently been learning about potential theory in the complex plane, and I would like to understand why the following proposition is true:
Suppose $f: \mathbb C \to \mathbb R$ is smooth, ...

**2**

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**0**answers

258 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 electrostatic ...

**1**

vote

**0**answers

37 views

### Uniquenss of domain with given interior newtonian potential

The newtonian potential of a domain $\Omega$ is defined by
$\Gamma*(\chi_{\Omega})$ ($\Gamma$ is the fundamental solution of laplacian operator $\Delta$), i.e. the convolution of indicator function ...

**1**

vote

**0**answers

53 views

### connectedness of coincidence set

Consider the following obstacle problem in the whole domain $\mathbb{R}^n$
min{$\Delta u$, $u$-$\phi$}=0
with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$ and $\phi$ (can be assumed ...

**1**

vote

**0**answers

109 views

### Does Newtonian capacity increase strictly when mass is spread?

We start with two disjoint compact sets A and B with positive capacities. Then, we translate B s.t. $B+rv$ is disjoint from A and B and ,more importantly, $dist(x,y)<dist(x,y+rv)$ for all $x\in A$ ...

**1**

vote

**0**answers

31 views

### Taking the potential of a super additive measure

In recent research of my coauthors and me, it has become necessary to consider the Riesz potential of a superadditive measure.
Recall that the $s$-dimensional Riesz potential of a finite Borel ...

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vote

**0**answers

36 views

### Harmonic Bergman spaces on graphs

Harmonic Bergman spaces on Euclidean domains are a set of harmonic functions on a domain that are from $L^{p}$ of that domain. I tried to find something on harmonic Bergman spaces on graphs because we ...

**1**

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**0**answers

258 views

### Strong minimum principle for maximal plurisubharmonic functions

Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...

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**0**answers

55 views

### Capacity of two disks

Is there an explicit formula for the (logarithmic) capacity of a union of two disjoint disks? As far as I understand, one can assume without loss of generality that the disks have the same radii (...

**0**

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20 views

### Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$.
Let $A \subset E$.
Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator.
I want to prove that $A$ is absorbing.
I know that it is ...

**0**

votes

**0**answers

71 views

### “increasing” the logarithmic energy of certain measures

Let $0<a<b<1$ and $f\in L^2[0,a]$ be a real-valued function with $\int_0^af^2=1.$ Define its logarithmic energy by $$\mathcal{E}_a(f)=\int_0^a\int_0^af(x)f(y)\log\frac{1}{|x-y|}dxdy$$
Q. ...

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22 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...

**0**

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88 views

### How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$

I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset \mathbb{C}^n$...

**0**

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39 views

### References for symmetric α-stable process (SSP) for $a>2$

Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...