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1
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2answers
59 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{ ...
8
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1answer
80 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
4
votes
1answer
77 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 ...
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1answer
47 views

Show that requiring Electrostatic potential to be a stationary point of Electrostatic potential energy is equivalent to Laplace's equation.

Suppose we want to find the electrostatic potential $\phi$(r) inside of some volume $V$ with known boundary conditions. The physical field configuration should minimize the electrostatic potential ...
0
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1answer
57 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 $$ ...
6
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0answers
246 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). ...
4
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0answers
396 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 ...
4
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0answers
110 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
77 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 simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
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} ...
3
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0answers
85 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
118 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 ...
2
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0answers
23 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
2
votes
0answers
126 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
71 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
80 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 ...
2
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0answers
51 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 ...
1
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0answers
27 views

Harmonic functions in the upper half plane

It is a cautionary remark that is often made that solutions to the Dirichlet problem (with continuous boundary conditions) are not unique when the domain in question is the upper half plane. Yes, you ...
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0answers
22 views

Newtonian potential of torus

Is there an explicit expression of the Newtonian potential for torus? As the expression for ball is clear by calculation. Definition of Newtonian potential of domain $\Omega$ at x is defined to be ...
1
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0answers
30 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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0answers
391 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
129 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
69 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
99 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 ...
1
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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
105 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
9 views

Energy functional of a uniform distribution of mass on a circle

Define $\nu$ as the measure in the Borel sets of $\mathbb{R}^{2}$obtained by uniformly distributing a unit mass on the circumference $\vert x \vert = 1$. Define its energy as $$I_{2}[\nu] = ...
0
votes
0answers
31 views

How to get between the equivalent definitions of Newtonian capacity?

Here are the two definitions: (1) $$\mathrm{Cap}(A)=\left[\inf \left\{\int\int |x-y|^{d-2}\mu(dx)\mu(dy):\mu \; \text{a probability measure on} \; A \right\}\right]^{-1}$$ and (2) ...
0
votes
0answers
57 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
44 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 ...
0
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0answers
30 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 ...