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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 (...
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### 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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### 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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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 $... 1answer 82 views ### Minimum of two plurisubharmonic functions I know that in general for$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$? I ... 1answer 102 views ### Riesz potential inequality Assume that$\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 $$\int_{\Omega} |x-y|^{\alpha-n} dy\... 1answer 60 views ### Seeking a specific proof of endpoint boundedness of Riesz potential The Riesz potential is defined by$$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$Once f\in L^{d/\alpha}(\mathbb{R}^n), then I_\alpha f(x)\in BMO. ... 1answer 103 views ### How to prove the Hölder continuity of a function u by evaluating \int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx? I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results: Let 0<\alpha<1, and B_1 be the unit ball centered at origin in \... 1answer 114 views ### Deriving Newtonian capacity of sphere from Brownian motion We have the following result by Spitzer (see (1) or Port) lim_{t\to \infty}\frac{1}{t}\int_{\mathbb{R}^{n}/B_{r_{0}}}P_{x}(T_{B_{r_{0}}}<t)dx=Cap(B_{r_{0}})=\frac{r_{0}}{4\pi} By Chuancun and ... 1answer 35 views ### P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty) imply Cap_{N}(A)<Cap_{N}(B), where Cap_{N} is Newtonian capacity We start a Brownian motion at x\in [B(0,r)]^{c}, where B(0,r) is a large enough ball that contains compact sets A and B. In other words, the B.M. starts on the exterior of A and B. Then ... 1answer 124 views ### Number of linear independent equations Is there any general rule to find the number of linearly independent equations such that$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$where$L_i$is a ... 1answer 69 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$u:V\to\mathbb{R}\cup\{-\infty\}$,$u(z)=\...
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 $\int_{\partial B} \xi(y)dS(y) = 0$ ...