Questions tagged [potential-theory]

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

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30 views

Average behavior of Green's function near the boundary

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for ...
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$u\in C_0^3(\Omega)$ satisfis $ -\Delta u=f(x), x\in\Omega. $Prove that $ \sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} dx\leq n\int_{\Omega}f^2 dx. $

Suppose $u\in C_0^3(\Omega)$ satisfis $$ -\Delta u=f(x), x\in\Omega. $$ Prove that $$ \sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} \mathrm{d}x\leq n\int_{\Omega}f^2\mathrm{d}x. $$ I have tried as ...
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How to prove potencies of matrices by induction

a) The matrix has the property ajk = 0 for 1≤j≤k≤N. Proof: A^N = 0. Note: use the induction over N. b) With the help of a), calculate the expression B^20 for the real matrix enter image description ...
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Potential flow around ellipse

I stuck at the following problem: Consider a planar incompressible potential flow (i.e., an irrotational flow) around an elliptic profile \begin{equation*} P = \lbrace (x,y) \in \mathbb{R}^2 | \frac{...
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Sum of subharmonic functions attaining a maximum in an open, connected set.

I want to prove the following statement: If $p_1,..,p_n$ are subharmonic functions on an open, connected subset $V$ of the complex plane and if $p_1+...+p_n$ attains a maximum on $V$, then each $p_i$ ...
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How to show that $\int_{B(x, t)} \Delta f d \lambda=t^{N-1} \int_{S} \frac{\partial}{\partial t} f(x+t y) d \sigma(y)$?

Let $f$ be a $C^{2}$ function on an open set which contains $\overline{B(x, r)} .$ We can use first Green's formula and then differentiation under the integral sign to obtain $$ \int_{B(x, t)} \Delta ...
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A series sum in a potential flow problem

A point source is located at the origin in a complex plane and two parallel walls are held at $\pm ib$. The complex potential should be solved by the method of image. The sum takes $$ w(z)=\frac{Q}{2\...
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On a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$, can we write the Laplace-Beltrami in the prescribed form on the whole space?

I have this one confusion. Let $G$ be a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$ . Then can we write its Laplace-Beltrami $L$ in the 'usual elliptic form' i.e. $$\sum_{i,j=1}...
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109 views

Harmonic function extension and Conformal maps

Let $\bigcup_{j=1}^n [0,a_i] = S$ be a 'star' where $a_i \in \mathbb{C}$ and the $[0,a_i]$ denote the line segments from $0$ to $a_i$ in the plane, all of the $a_i$ here are distinct and nonzero. ...
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Extract a potential part of a function

Given a function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$, expressed in spherical coordinates, suppose I want to decompose $f$ into a potential piece and a non-potential piece, $f = f_P + f_{NP}$ and ...
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38 views

Positive function which is harmonic in every compact set in a cone

Let $C \subset \mathbb R^3$ be an open unbounded cone with vertex at the origin (I do not care much about the angle of the cone but assume that $C \cap \partial B(0,1) = B((0,0,1), r) \cap \partial B(...
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Greens function with pole at infinity - Ahlfors Conformal Invariants chapter 2

In the textbook 'Conformal Invariants : Topics in Geometric Function Theory' page 25, there is the following formula (highlighted in yellow): I am very stuck on how Ahlfors manages to get formula (2-...
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$p$-admissible weight, which is not an $A_p$ weight

Let $\Omega\subset\mathbb{R}^N$ be a bounded and smooth domain with $N\geq 2$. Let us consider the following class of weights $$ B_s=\{w\in W_p: w^{-s}\in L^1(\Omega)\text{ for some }s\in Y\},\quad 1&...
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Riesz potentials of $\mathscr{L}^p_\beta\left(\mathbb{R}^d\right)$ functions

In the Chapter V of Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30. Princeton, N.J.: Princeton University Press. XIV, 287 p. (1970)....
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132 views

Reverse Hardy-Litllewood-Sobolev inequality

Some time ago, I asked here: How to use Hardy-Littlewood-Sobolev inequality to estimate an integral involving two fuctions and Riesz Potential. about estimates involving the Hardy-Littlewood-Sobolev ...
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1answer
70 views

Taking the Vector Gradient of a Function in $(r)$ where $r = |\mathbf{\vec{r}}|$

I have tried looking for the answer to this online, but probably due to my personal lack of knowledge have been unable to ask the question effectively. Essentially I would like to understand how to ...
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How can I solve this PDE $h(\phi_{xx}+\phi_{yy})+h_{x}\phi_{x}+h_{y}\phi_{y}=0, h=5-e^{-x^4-y^2}$

$\phi(x,y)$ is the velocity potential ($\frac{\partial \phi}{\partial x}=-u,\frac{\partial \phi}{\partial y}=-v$) for a steady flow in a 2D ocean. $h(x,y)=5-e^{-x^4-y^2}$ is the water depth at $(x,y)$....
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Can "abstract potential theory" be used to establish maximum principles for more general PDOs?

As I understand it, this paper uses ideas from "abstract potential theory" (harmonic spaces, sheaves etc.) to find a maximum principle applying to unbounded domains. But its assumptions on ...
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Extension of superharmonic functions

In the "Classical Potential Theory" of Armitage and Gardiner, the following result is stated as an exercise (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open ...
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Riesz decompostion (Subharmonic function)

I was reading a proof of the Riesz decomposition theorem and I was not able to understand how the author has arrived at the given equation for $L(\mathcal{w};0,r)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{...
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Logarithmic Potential [closed]

I was studying logarithmic potential $f(z)=\int_E \log(\frac{1}{ \mid{z-a} \mid })d\mu(a)$ (Tsuji's Potential theory in modern function theory ). I am trying to prove $f(z)$ is harmonic outside of $E$,...
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Riesz measure of a subharmonic function and that of its regularization

Suppose $u$ is a subharmonic function on a bounded open set $V$ of $\mathbb{R}^m$ ($m\geq2$). Let $u_n$ be a sequence of smooth subharmonic functions that decreases pointwise to $u$ (obtained by ...
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Hitting probability and recurrent Brownian motion

Consider a Brownian particle $B(t)$ in $\mathbb{R}^2$ starting at the origin, and let $U$ be an open set in $\mathbb{R}^2$. For a positive real number $\tau$, define $$ P_1(\tau) = \text{ probability ...
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Question about non-negative harmonic polynomial on cone, in relation to derivation of GUE's Ginibre formula.

Let $\mathbb{R}_{>}^n$ be the cone $\{x\in \mathbb{R}^n|\,\forall j \in \{1,...,n-1\}:\, x_{j+1}<x_j\}$. Let $p:\mathbb{R}^n \to \mathbb{R}$ be a (non-zero) homogeneous harmonic polynomial of ...
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1answer
45 views

Differential equation for a vector potential

From Helmholtz’s theorem, any smooth vector field $\mathbf{F}$ that goes to zero at infinite distance can be uniquely decomposed everywhere in the sum of a divergence free component and an ...
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Green's function: Proving g(a,b)=g(b,a)

I was reading a proof of g(a,b)=g(b,a) for Green's function. There is an integral (equation 1) M Tsuji's Potential theory in modern function theory chap 1 proving Greens function is symmetric that I ...
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Locally positive capacity on manifold with boundary

Let $M$ be a (complete) Riemannian manifold with boundary $\partial M$ and with $\mu$ being the Riemannian measure on $M$. Given a compact set $K$ in $M$ and $\Omega$ being a precompact open set in $M$...
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What is the norm of the single and double-layer potential operators in $\mathcal{L}(L^2(\partial \Omega),H^{1}(\partial \Omega))$.

Let $S$ the following single layer potential associated to the laplacian problem : $$S[\varphi](x):= \int_{\partial \Omega} G(x-y) \varphi(y) d\sigma(y) , \quad \varphi \in L^2(\partial \Omega), \quad ...
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Minimization of a classical energy functional (Gibbs measure)

Let $\mathcal{P}(\mathbb{R}^2)$ be the space of probability densities on $\mathbb{R}^2$. Consider the following energy functional composed of entropy, a potential energy $U$ which is uniformly convex, ...
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Laplace equation for circular symmetry: given a potential $\phi (z,0)$, how to find $\phi (0,r)$?

The potential function $\phi(z,r)$ has circular symmetry with respect to $r$. It also satisfies the Laplace equation $\partial \phi^2/\partial z^2 + \partial \phi^2/\partial r^2 + (1/r)\partial \phi/\...
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L2 Dirichlet flow of the linear Fokker-Planck equation

I can see how the heat equation (say on $\mathbb{R}^d$) : $\partial_t \rho=\Delta\rho$, can be viewed as a $L_2$ gradient flow of the energy $$ E(\rho)=\frac{1}{2}\int_{\mathbb{R}^d} |\nabla \rho(x)|^...
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Boundary characterization of harmonic functions on $\Bbb R^n$

I was reading some paper on Martin/Possion boundaries, but most of them concerned some kind of bounded cases. So I wondered, is there any boundary-like characterization of a kind there exists an $(n-1)...
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Physical intuition for the representation formula for harmonic functions

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with $C^{1}$ boundary, $u$ is harmonic in $\Omega$ and belongs to $C^{1}(\bar{\Omega})$. Then by Green's formula we have for every $x \in \Omega$...
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1answer
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How to use Hardy-Littlewood-Sobolev inequality to estimate an integral involving two fuctions and Riesz Potential.

Recently I've been studying some PDEs involving Riesz potential and I saw the following assertion: If $u,v \in H^{1}(\mathbb{R}^{2})$, then $$\int_{\mathbb{R}^{2}}(I_{\beta} \ast |u|^{\frac{\beta}{2}+...
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1answer
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Helmholtz equation Inequality (potential theory)

I was reading the chapter about surface potential. I came across an inequality $$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|,$$ $k$ is a complex number here. Since $k$ is a complex number here, I am ...
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1answer
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Let $u$ be a negative subharmonic function on $D(0,1)$. Then $\limsup_{r\to 1^-} \frac{u(rz_0)}{1-r}< 0$ for each $z_0\in \partial D(0,1)$.

This exercise has a hints to follow which is apply maximum principle $f(z)=u(z)+c\log|z|$ on the domain $\frac{1}{2}<|z|<1$. Now if I choose $c>0$ then since $u<0$ in the whole domain so ...
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1answer
102 views

Double layer potential in 1d?

I would like to illustrate the double layer potential idea with a simple 1d example, but seem to run into a situation where the resulting integral equation is singular. The problem is $u''(x) = 0$ on $...
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1answer
51 views

A subharmonic function in a ball

This question is related to this (A nonnegative harmonic function in a ball). Let $B(a,r)$ be an open ball in $\mathbb{R}^m$, ($m\geq2$). Is there a function $u$, $u\not\equiv0$, that is subharmonic ...
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A superharmonic function that is zero on a compact set 2

This question is related to this (A superharmonic function that is zero on a compact set) Let $K$ be a compact subset of $\mathbb{R}^m$ with $m\geq 3$, and $\Omega=\mathbb{R}^m\setminus K$. Is there a ...
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continuity of $J(r,y) = |S^{n-1}|^{-1} \int_{\mathbb{R}^n} |r \omega - y|^{2-n} \text{d} \omega$

In a proof I'm trying to understand the continuity of this function is just being assumed and I'm a little stuck trying to show why. This function is defined on $ \mathbb{R}^{n+1} $ with r the radius ...
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Regularity of continuous function on Berkovich spaces: is there an example of a continuous function which is not of bounded differential variation?

I'm reading Baker and Rumley's Potential theory and Dynamics on the Berkovich Projective Line, and am trying to understand their definition of functions of bounded differential variation. Given a ...
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1answer
45 views

A superharmonic function that is zero on a compact set

Let $K$ be a compact subset of $\mathbb{R}^m$ with $m\geq 3$, and $\Omega=\mathbb{R}^m\setminus K$. Is there a continuous, positive superharmonic function $u$ (or at least just superharmonic) on $\...
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1answer
148 views

Question about continuity of stopping time probability in proof of Dirichlet problem in Stein and Shakarchi

In Chapter 6 in the book Functional analysis by Stein and Shakarchi the following theorem (Dirichlet problem) is proved: $\mathcal{R}$ denotes a bounded open set and a point $y$ is called regular if $...
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3-dimensional vector potentials

Is there a vector potential of 3-dimensinal field? A vector potential $\boldsymbol A(\boldsymbol x),\ \boldsymbol u(\boldsymbol x)=\nabla\times\boldsymbol A(\boldsymbol x)\ $is $$ \boldsymbol A(\...
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29 views

The maximum principle for subharmonic functions on unbounded open sets

Let $K$ be a compact set of $\mathbb{R}^m$ and let $m\geq2$. Suppose $u$ is subharmonic on $V=\mathbb{R}^m\setminus K$ and $$\limsup u(x) \leq0\hspace{1.5cm} (1) $$ when $x\to y$ from inside $V$, ...
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How can I solve this poisson equation numerically?

I am trying to solve a potential flow field of a flow around a circle using a method called the Boundary Data Immersion Method, which immerses the boundary conditions into the fluid using a smoothing ...
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43 views

References for magnetic Sobolev spaces

I am working on PDE. Recently I have been studying magnetic Sobolev spaces. While the theory is clear to me, having very little knowledge about physics, I have almost no idea how these spaces help ...
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Potential theory and examples of hyperharmonic functions

This follows the book "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians" by Bonfiglioli, Lanconelli, and Uguzzoni. In chapter 6, the book introduces abstract harmonic ...
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1answer
146 views

How to prove case 1 of Theorem 4.3 in "Calculus of Several Variables" by Serge Lang?

I am reading "Calculus of Several Variables 3rd Edition" by Serge Lang. How to prove case 1 of Theorem 4.3 in "Calculus of Several Variables" by Serge Lang? In Theorem 4.1, for any ...
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1answer
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Do we really need to use the assumption to show that $\phi$ is well defined. (Serge Lang Calculus of Several Variables)

I am reading "Calculus of Several Variables 3rd Edition" by Serge Lang. Do we really need to use the assumption to show that $\phi$ is well defined.

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