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2
votes
2answers
179 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 $ ...
0
votes
0answers
18 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
0answers
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. ...
2
votes
0answers
158 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 ...
2
votes
0answers
121 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 ...
4
votes
3answers
262 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 ...
6
votes
0answers
382 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 ...
0
votes
0answers
19 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 ...
3
votes
0answers
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 ...
24
votes
1answer
490 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 ...
0
votes
1answer
164 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 ...
1
vote
1answer
58 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 $I_\alpha f(x)\in BMO$. ...
1
vote
1answer
100 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 ...
4
votes
0answers
105 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 ...
2
votes
0answers
38 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 ...
1
vote
1answer
35 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 $A$ and $B$. In other words, the B.M. starts on the exterior of $A$ and $B$. Then ...
6
votes
1answer
307 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 ...
1
vote
2answers
96 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
0answers
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 ...
3
votes
3answers
335 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 ...
8
votes
1answer
199 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
1answer
68 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
1answer
139 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 ...
5
votes
1answer
159 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 ...
-1
votes
1answer
108 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 ...
3
votes
2answers
253 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 ...
0
votes
0answers
38 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 $\alpha>2$ ...
2
votes
1answer
135 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
0answers
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
0answers
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
0answers
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
1answer
113 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 ...
0
votes
2answers
96 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
vote
0answers
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 ...
0
votes
1answer
158 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
1answer
81 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
0answers
67 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
0answers
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 ...
3
votes
1answer
201 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 ...
0
votes
1answer
121 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
1answer
247 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 ...
1
vote
0answers
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
vote
1answer
100 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
1answer
283 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 ...
20
votes
1answer
1k 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 ...
2
votes
1answer
277 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 ...
8
votes
1answer
569 views

How does electric potential relate to mean curvature?

Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$. Now consider the electric potential generated by this uniform mass ...
1
vote
1answer
124 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
1answer
68 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 $u:V\to\mathbb{R}\cup\{-\infty\}$, ...
2
votes
0answers
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, ...