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24
votes
1answer
465 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 ...
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 ...
12
votes
2answers
1k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
10
votes
1answer
351 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 ...
8
votes
1answer
183 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 ...
8
votes
1answer
563 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 ...
8
votes
0answers
110 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 ...
7
votes
2answers
546 views

Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator. We wish to solve the Dirichlet problem ...
6
votes
1answer
301 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 ...
5
votes
2answers
5k views

Version of the Poincaré Inequality

Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality $$\|u\|_p \le C \|\nabla u\|_p$$ ($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used ...
5
votes
2answers
697 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 ...
5
votes
1answer
470 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 ...
5
votes
1answer
565 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
1answer
155 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 ...
5
votes
0answers
174 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
votes
2answers
197 views

An extremal type problem on segments

I am interested in the following extremal-type problem. Let us define $\Psi$ as $$\Psi(x)=\max_{f\in L^2[0,x] \,\,with\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$ on ...
4
votes
0answers
104 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
votes
0answers
336 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 ...
4
votes
0answers
911 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 ...
3
votes
3answers
327 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 ...
3
votes
1answer
222 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
1answer
193 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 ...
3
votes
2answers
248 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 ...
3
votes
0answers
190 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
votes
0answers
386 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 ...
3
votes
0answers
276 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 ...
2
votes
1answer
496 views

Compact sets of the complex plane having the K-property ?

I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples. As it is ...
2
votes
1answer
446 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, ...
2
votes
1answer
124 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 ...
2
votes
1answer
251 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 ...
2
votes
1answer
280 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
0answers
37 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
votes
0answers
65 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
175 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
0answers
103 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
votes
0answers
250 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
1answer
63 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 ...
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 ...
1
vote
1answer
75 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 ...
1
vote
1answer
93 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} ...
1
vote
1answer
48 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
94 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 ...
1
vote
1answer
109 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 ...
1
vote
1answer
33 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 ...
1
vote
1answer
122 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
65 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
1answer
195 views

A variational problem involving a negative fractional Soboblev norm.

I've run into the problem of trying to evaluate the following: $\max_{ \xi} \iint_{\partial B \times \partial B} \xi(x) \Phi(|x-y|) \xi(y) dS(x)dS(y)$ subject to $\int_{\partial B} \xi(y)dS(y) = 0$ ...
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
51 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
103 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 $x\in A$ ...