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2answers
34 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
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1answer
43 views

Volume potential (show continuity)

Consider $$ \Omega:=B_{R_2}(0)\setminus \overline{B_{R_1}}(0)\subset\mathbb{R}^3 $$ with $0<R_1<r_2<\infty$ (hollow ball) and $$ ...
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0answers
208 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
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0answers
101 views

Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
3
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0answers
57 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=inf\int_E\int_E log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $inf$ is taken over all ...
3
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0answers
224 views

Finding the Green Function of the upper half ball

Find the Green function of $\Omega:=\left\{x\in\mathbb{R}^n:\lVert x\rVert<R, x_n>0\right\}$ and show that the function you've found is indeed a Green function! You are allowed to use ...
3
votes
0answers
41 views

Bounds on solutions of elliptic pde's in the whole plane

My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} ...
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0answers
71 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
3
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0answers
103 views

A finely open set, not open up to polar set?

Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, which is not open up to a polar set (i.e. zero capacity), i.e., there does ...
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0answers
78 views

Potentials in Probability Theory

Could someone give an intuitive interpretation of potentials in the field of probability theory. How do they link to the theory of stochastic processes. And maybe link this with SEP. References are ...
2
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0answers
59 views

Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
2
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0answers
66 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
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0answers
49 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
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0answers
74 views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
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0answers
46 views

Green function of the first quadrant

Find the Green function of the first quadrant $x_1>0, x_2>0$. HINT: Use the Green function of the half space $\Omega:=\left\{x\in\mathbb{R}^n : x_n > 0\right\}$ which is given by ...
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0answers
65 views

Estimation (Newton potential)

Let $f$ be a $C^2$-function on $\mathbb{R}^n$ with compact support. Let $N$ be the Newtonpotential on $\mathbb{R}^n$ and $u:=N\star f$ (i.e. a solution of the potential equation $\Delta u=f$). ...
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0answers
87 views

discrete harmonic extension (an exercise of Grimmett's “probability on graphs”)

I'm struggling with exercise 1.3 in Grimmett's book "probability on graphs". Take $G = (V,E)$ a finite connected graph with given positive conductances $(w_e)_{e \in E}$, and let $(x_v)_{v \in V}$ be ...
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0answers
59 views

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

Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is the following true ? (Question 1) Let $p(z,t) = \frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ ...
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0answers
96 views

Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$

Show that $$ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx = \log{\frac{|z+\sqrt{z^2-1}|}{2}},\quad z \in \mathbb{C} $$ How can I apply the Joukowski conformal map to this problem? Thanks.
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0answers
19 views

Help interpret Sommerfeld radiation condition.

I am studying the Sommerfeld radiation condition. In potential theory, a solution $u(r,\theta)$ to a partial differential (such as the Helmholtz equation $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
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0answers
35 views

Dirichlet boundary problem and volume potential

Consider the Dirichlet boundary value problem for the Laplace-equation (1) $-\Delta u=f$ in $\Omega:=B_R(0):=\left\{x\in\mathbb{R}^3: \lVert x\rVert < R\right\}$ (2) $u=0$ on ...
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0answers
23 views

Two more little tasks concerning Dirichlet boundary value problem

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap ...
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0answers
28 views

Finding 2D potential distribution given boundary conditions.

I came across a certain problem which I don't know how to solve. I'm looking for a few hints or suggested readings which would lead me to a solution. The problem is to solve for the potential ...
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0answers
101 views

Properties of Subharmonic functions 1

There are 3 theorems in my textbook (My textbook doesn't have a solution): Why do we have ${\color{Red} (I)}$. $1/$ Let $\Omega$ be an open subset of $\mathbb{C}$. If $f : \Omega \to ...
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votes
0answers
18 views

Cartan inequality, Polya inequality for analytic functions.

I have a real analytic function $F$ and I don't want to worry about radius of convergence. I saw in a paper that Cartan's inequality implies, if $|F|<1 $ and there is an $ |x |<1/2$ so that $|F ...