# Questions tagged [potential-theory]

The potential-theory tag has no usage guidance.

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### 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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121 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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63 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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55 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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86 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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133 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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107 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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105 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 ...

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75 views

### Green's third identity potential massive object

Consider a massive object occupying a volume $U$ with boundary $\partial U$. Let the gravitational potential inside be $V_{in}$ and outside $V_{out}$
Normally the gravitational field of a massive ...

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102 views

### A variant of the Hardy-Littlewood-Sobolev inequality in one dimensional case

The classical one-dimensional case is: for $\alpha\in\langle0,1\rangle$ and $1<p<q<\infty$ such that $1/q=1/p-\alpha$, there is a constant $C_p>0$ depending on $p$ such that
$$\|I_\alpha f\...

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64 views

### criterions for polar set of Feller processes

Suppose $X_t$ is the solution to
$$
d X_t=b(X_t)dt+dL_t,\quad X_0=x.
$$
where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.
Assume $\Gamma\subseteq ...

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150 views

### Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...

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163 views

### Capacity of a unit disk with a small bump

Let $A_r = \{z\in\mathbb{C}: |z|\leq 1\}\cup\{z\in\mathbb{C}: |z-1|\leq r\}$ be the unit disk with a small "bump" (I'm interested in the regime $r\to 0$). What can be said about the logarithmic ...

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443 views

### Calculation of logarithmic capacity?

I am reading this paper about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like ...

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### Defining a capacity wrt. positivity preserving forms that are not regular?

Let $(X,m)$ be a locally compact measure space countable at infinity. Suppose we have a bilinear form $a:H \times H \to \mathbb{R}$ on a Hilbert space $H \subset L^2(X)$.
The form is coercive and ...

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48 views

### Dirichlet PWB solution convergence when removing a shrinking disk

This was asked with bounty in math.SE without success.
Let $D \subset \Bbb R^2$ be a bounded domain, $h \in \mathcal C^0(\partial D)$, let $\tilde h = \sup \mathcal F_h$ be the Perron solution of ...

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102 views

### Two questions related to Dirichlet spaces and Sobolev spaces

I want to ask a question that arises from reading this paper.
Let $X$ be a locally compact space which is countable at infinity and let $\xi$ be a Radon measure on $X$. Suppose $V$ is a Hilbert ...

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115 views

### Optimal Holder regularity estimates for the single layer potential

I need a statement that must be classical but it's being very difficult (even after consulting experts) to find a reference. Let's try here!
Let $f$ be a function on the boundary of a bounded domain $...

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230 views

### References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology

Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...

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43 views

### When is a 2D homogenous potential essentially self-adjoint? What about the potential $V(x,y)=x^4+y^4-\lambda x^2y^2$?

Suppose I consider the operator
$$
-\Delta+V$$
for some potential $V(x)$ for $(x,y)\in\mathbb{R}^2$, as the closure of the corresponding operator on smooth compactly supported functions. If I assume ...

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96 views

### Charge density in toroidal coordinates

I am trying to compute the volume charge density present in a toroidal conductor. For a current in the azimuthal direction, the potential inside the toroid is:
$$\Phi(\eta>\eta_{0},\xi,\Theta)=A+\...

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189 views

### How to choose contour for rational approximation

Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\...

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76 views

### A pluripolar set that disconnects an open?

Let $\Omega$ be a bounded open set of $\mathbb{C}^{n}$, and $F$ a pluripolar subset of $\mathbb{C}^{n}$. We know that if $F$ is closed, then $\Omega\setminus F$ is a connected open set. What if $F$ is ...

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107 views

### Compactness of double layer potential operator for general elliptic equations

Given is a bounded domain $\Omega$. Let's denote by $\Gamma$ the fundamental solution of the Laplacian equation. Then for a function $\psi$ defined on $\partial\Omega$, one can define the double layer ...

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169 views

### Is this a superharmonic function?

Hi everyone: Let $ \Omega $ be a bounded open set of $ \mathbb{R}^{N} $, $ N\geq2 $, and $ F\subset \Omega $ with empty interior. Suppose there exists a superharmonic function $ u $ on $ \Omega\...

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438 views

### Harmonic measure

Hi everyone: Let $ \omega $ be a bounded open set in $ \mathbb{R}^{q} $, $ q\geq 2 $, and $ E $ a subset of the boundary $ \partial\omega $ that has harmonic measure zero in $ \omega $. Let $ V $ be ...

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334 views

### Canonical English edition of Dellacherie and Meyer's “Probabilities and Potential”

Probabilities and Potential by Dellacherie and Meyer is a "bible" of probabilistic potential theory, Markov processes, and many related topics. I want my library to acquire it, but I am a bit ...

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### 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 (...

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### “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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145 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}\...

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383 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 $ [...

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60 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 \,...

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243 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 ...

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90 views

### Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by
$$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...

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180 views

### $\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...

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120 views

### How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan.
In his lecture, he uses the following results:
Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in $\...

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274 views

### An extremal type problem on segments

I am interested in the following extremal-type problem.
Let us define $\Psi$ by
$$\Psi(x)=\max_{f\in L^2[0,x] \,\,\text{with}\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$
on $(0,\...

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126 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 ...

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333 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

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750 views

### A problem of potential theory arising in biology

Let $K_0$ and $K_1$ be two bounded, disjoint convex sets in $R^n,n\geq 3$,
and
$u$ the equibrium potential, that is the
harmonic functon in $R^n\backslash\{ K_0\cup K_1\}$ such that $u$
has boundary ...

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111 views

### Is zero a regular point for a drifted $\alpha$-stable process?

We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$,
where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha
\in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$,
and $...

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103 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 $\Omega\subset \mathbb{C}^n$...

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231 views

### Which domain maximizes the energy of the Lebesgue measure?

This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...

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106 views

### Oscillation of subharmonic functions of slow growth

Given a sequence of real numbers $c_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z_k\to\infty$ with $|z_k|<k$ such that
$$\displaystyle\...

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161 views

### Positivity of logarithmic energy of certain measures

Let $\Gamma$ be a smooth closed curve in the complex plane (for all practical purposes). Assume $f$ is a real-valued continuous function defined on $\Gamma$ and let $d\mu=fdm$, where $dm$ is the ...

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53 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 $\alpha>2$ ...

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211 views

### Approximation of subharmonic functions

Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi_\epsilon$ be a standard smoothing kernel, namely
$$\chi_\epsilon(x)=\frac{c_n}{\...

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### 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 ...

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64 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 ...

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133 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 $x\in A$ ...