The potential-theory tag has no usage guidance.

**7**

votes

**1**answer

281 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 ...

**23**

votes

**0**answers

377 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 ...

**1**

vote

**2**answers

85 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 ...

**0**

votes

**0**answers

85 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 ...

**8**

votes

**1**answer

180 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 ...

**1**

vote

**1**answer

60 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
...

**0**

votes

**1**answer

131 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 ...

**0**

votes

**0**answers

34 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 $\alpha>2$ ...

**0**

votes

**0**answers

27 views

### Uniqueness of homogeneous second kind Fredholm equation

I have the following equation:
$-\frac{1}{2}\phi + W\phi - V(f\phi) = 0 $
and I intend to prove uniqueness of the solution in a appropriate Sobolev space. V and W denote the direct values of the ...

**0**

votes

**0**answers

19 views

### Connectedness of conincidence set [duplicate]

Is there any criterion for connectedness of coincidence set, for obstacle question
$min{Δu, u-ϕ}=0$ and with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$?
Or any other kinds of ...

**2**

votes

**1**answer

114 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
...

**1**

vote

**0**answers

33 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

49 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

95 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

**1**answer

106 views

### Deriving Newtonian capacity of sphere from Brownian motion

We have the following result by Spitzer (see (1) or Port)
$lim_{t\to \infty}\frac{1}{t}\int_{\mathbb{R}^{n}/B_{r_{0}}}P_{x}(T_{B_{r_{0}}}<t)dx=Cap(B_{r_{0}})=\frac{r_{0}}{4\pi}$
By Chuancun and ...

**3**

votes

**0**answers

491 views

### Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tseva

I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer.
Let $G(t,x)$ be the fundamental ...

**1**

vote

**0**answers

27 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 ...

**0**

votes

**1**answer

19 views

### $P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity

We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets A B. In other words, the B.M. starts on the exterior of A and B.
Then if the ...

**0**

votes

**2**answers

79 views

### Newtonian capacity of sphere equals its hitting probability by Brownian motion?

Do we have $Cap_{N}(S(0,r))=P_{x}(T_{S(0,r)}<\infty)$ for $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a ball centered at the origin ?
I know for $x=0$, they are both equal to 1. How can I go about ...

**-1**

votes

**1**answer

103 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 ...

**1**

vote

**0**answers

26 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 ...

**5**

votes

**1**answer

147 views

### Convergence in energy of bounded (semi)subharmonic functions

Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...

**0**

votes

**1**answer

141 views

### Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials.
If $\Gamma(t,x)$ is the fundamental ...

**1**

vote

**1**answer

74 views

### Minimum of two plurisubharmonic functions

I know that in general for $u,v\in PSH$ (plurisubharmonic) $\min\{u,v\}$ is not a $PSH$ function. Are there any known results under which conditions on $u$,$v$ a function $\min\{u,v\}$ is $PSH$?
I ...

**2**

votes

**0**answers

61 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**

votes

**0**answers

164 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 ...

**0**

votes

**1**answer

103 views

### uniqueness for Poisson equation in R^d with mildly regular data

I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...

**3**

votes

**1**answer

195 views

### Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$
$$
f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...

**3**

votes

**1**answer

183 views

### reference request: Riesz/Newton potential and HLS inequality in L1.logL1

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$
$$
f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...

**1**

vote

**0**answers

35 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**

vote

**1**answer

86 views

### Riesz potential inequality

Assume that $\Omega$ is a domain in $\mathbf{R}^n$ with the same area as a ball $B(x,r)$ and let $\alpha\in[1,0)$. I need the reference for the following inequality $$\int_{\Omega} |x-y|^{\alpha-n} ...

**2**

votes

**1**answer

217 views

### Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...

**3**

votes

**3**answers

325 views

### Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...

**20**

votes

**1**answer

891 views

### The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...

**1**

vote

**1**answer

117 views

### Number of linear independent equations

Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a ...

**1**

vote

**1**answer

57 views

### A formula for the potential part in Riesz decomposition of simple subharmonic functions?

Consider a compact subset $E\subset\mathbb{C}$, holomorphic functions $f_j:V\to \mathbb{C}$, $1\leq j\leq k$, defined in a neighbourhood $V$ of $E$, and set $u:V\to\mathbb{R}\cup\{-\infty\}$, ...

**1**

vote

**0**answers

99 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, ...

**8**

votes

**0**answers

109 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 ...

**2**

votes

**1**answer

279 views

### Uniqueness of classical solution with degenerate boundary

Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial data
$u(x,0) = x$ for ...

**2**

votes

**0**answers

241 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 ...

**5**

votes

**0**answers

171 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 ...

**0**

votes

**1**answer

139 views

### Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: ...

**10**

votes

**1**answer

347 views

### Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...

**5**

votes

**1**answer

434 views

### Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...

**0**

votes

**1**answer

150 views

### On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...

**1**

vote

**1**answer

444 views

### Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...

**3**

votes

**2**answers

247 views

### Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...

**5**

votes

**1**answer

484 views

### Logarithmic potential

Given a continuous, compactly supported function $f$ on $R^2$, it is known that the logarithmic potential of $f$, that is
$$
U_{f}(x):=-\frac{1}{2\pi}\int\log|x-y|f(y)dy
$$
has the following decay at ...

**5**

votes

**2**answers

674 views

### Kähler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are ...

**1**

vote

**0**answers

250 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 ...