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2answers
33 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{ ...
0
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1answer
20 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
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1answer
27 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
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1answer
27 views

Some properties of capacity

Let $\Omega\subset\mathbb{R}^N$. For $K\subset \Omega$ we can define the $p$-capacity, $p\in (1,\infty)$ as the number $$\operatorname{cap}_p(K)=\inf \int_\Omega |\nabla u|^p$$ where the infimum is ...
1
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1answer
28 views

Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
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0answers
70 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
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 $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
3
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0answers
56 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 ...
8
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1answer
170 views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
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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 ...
3
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0answers
196 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 ...
2
votes
2answers
55 views

Assumption on $u$ harmonic in disc making $u$ unique with particular (discontinuous) boundary data

From an old qualifier: Let $$J_1 = \{e^{i\theta}: 0 < \theta < \frac\pi2\}, \,\,J_2 = \{e^{i\theta}:\frac\pi2<\theta< \pi\},\,\, J_3 = \{e^{i\theta}:\pi<\theta < 2\pi\}$$ be ...
1
vote
1answer
56 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
1
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2answers
74 views

Line integral over a vector field using a potential function.

Let $X(t)=\left ( \pi\cos t, t, e^{\sin t} \right ), 0\leq t\leq \pi $. And let $\textbf{F}(x,y,z)=\left ( 2xy+\frac{-y}{x^2+y^2}, x^2+ze^{yz}+\frac x{x^2+y^2}, ye^{yz} \right )$ be a vector field. I ...
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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 ...
0
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1answer
34 views

Two 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 ...
1
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0answers
41 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 ...
2
votes
1answer
102 views

Find the Green function of the upper half ball by using the Green function of the whole ball

Find the Green function of the upper half ball $\Omega:=\left\{x\in\mathbb{R}^n|\lVert x\rVert<R, x_n>0\right\}$ (for the Dirichlet boundary value problem of the Laplace equation). Show ...
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1answer
32 views

What is the consequence for the dyson sphere? [closed]

as you can see here Normal derivation the volume potential is constant in the origin and inside the hollow ball. What consequences does this have for a dyson-sphere? I do not know, because I ...
1
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1answer
47 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
0
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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 $$ ...
1
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1answer
72 views

Show that the Kelvin transformation is a diffeomorphism and find the inverse

Let $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. We call $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ with $$ y=\phi(x):=\frac{R^2}{\lVert ...
4
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1answer
81 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
2
votes
1answer
48 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
2
votes
1answer
39 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
3
votes
1answer
124 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...
0
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0answers
27 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 ...
0
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1answer
25 views

Understanding Lyapunov boundaries and where are they used?

Reading some potential theory, my book almost always uses a rather strong regularity condition on the boundaries to be of class $C^2$. My book refers to a slightly weaker case of Lyapunov boundaries. ...
1
vote
1answer
44 views

Prove that $u \circ f $ is plurisubharmonic on $\Omega_1$

I'm trying to show that the theorem in my book: Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on ...
1
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1answer
38 views

Show that $\widetilde{u} \in PSH(\Omega)$

EDITED I'm reading the book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992). I don't understand ...
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0answers
99 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 ...
2
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0answers
77 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 ...
0
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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 ...
2
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0answers
58 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 ...
3
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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} ...
0
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2answers
82 views

Prove that $\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |)) \in MPSH(\Omega)$

This's an example: For $u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where $z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} ...
1
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1answer
50 views

aproximation in Sobolev Spaces

consider $r>0 , p>1$ and $K \subset B(x_0 , 2r) \subset R^n$ . $K$ compact. Define the sets : $$A = \{ u \in C^{\infty}_{0} (B(x_0 , 2r)); \textit{ such that } \ u=1 \textit{ in a open ...
3
votes
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 ...
1
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1answer
72 views

Question which involves Riesz potential

Let $f :S^{n-1}\longrightarrow R_+$ , $P_{j,k}. j,k \in N$ be a spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics) and $\displaystyle{f\left(\frac{x}{|x|}\right)|x|^{1-\frac ...
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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 $\Delta u=f$). ...
3
votes
1answer
53 views

$f\in L^2(\mathbb{R}^3)$ implies $v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy\in W^{2,2}$?

Let $f\in L^2(\mathbb{R}^3)$ be a function with compact support and define $v:\mathbb{R}^3\to\mathbb{R}$ by $$v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy$$ Is true that $v\in W^{2,2}(\mathbb{R}^3)$ ...
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 ...
2
votes
1answer
62 views

Irrotational fields and change of reference frame

Given a reference frame $(x_1,x_2,x_3)$ and a vector field $\overrightarrow{V}(x_1,x_2,x_3)$, in this frame, if $\overrightarrow{\nabla}\times\overrightarrow{ V}(x_1,x_2,x_3)=0$ the field is ...
1
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0answers
84 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 ...
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 ...
0
votes
1answer
134 views

Problem on Yukawa Potential

One definition of the Yukawa potential on $R^n$ is the solution $G$ in the sense of distributions to $(-\Delta + \mu^2)G = \delta$. This 'green's function' is given by \begin{align*} G(x) = ...
6
votes
1answer
261 views

Green's function in 2D

How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m ...
3
votes
1answer
95 views

A reference to study p-admissible functions

I am studying p-admissible functions. I am using the book of heinonen (nonlinear potential theory of degenerate elliptic equations). I am searching for a good proof of the result: Suppose that w ...
3
votes
1answer
36 views

Determining the effective coefficient in a boundary value problem.

I'm trying to understand the setup given in Multiscale analytical solutions and homogenization of n-dimensional generalized elliptic equations, pg 41: Given the boundary value problem: ...
1
vote
1answer
174 views

Discontinuity of double-layer potentials

I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...