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.
382 questions
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Magnetic Field of Cylinder with Uniform Current Density
Though this is a problem from physics, I think it is more about mathematics.
Suppose an infinite cylinder $V$ of radius $a$ with its center aligned in $z$ axis carrying a current density
$$
\...
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Is there a polar finely open set?
The fine topology of the potential theory is the coarsest topology that makes superharmonic functions continuous in the extended sense. We know that $\mathbb{R}^n$ endowed with the fine topology is ...
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Series of potentials in box trap wave-function
I have a to find the wave-function of the following problem. I have free spineless fermions in a 2D box trap of size $L_x\times L_y$. To such problem, the solution is known and is
$$\Psi(x,y) =\sqrt{\...
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Electrostatic potential formula
The electrostatic potential $\phi\left(\boldsymbol{r}\right)$ is given as
$$
\phi\left(\boldsymbol{r}\right) = \frac{1}{4\pi\varepsilon_0}\int\limits_{V'}\frac{\varrho\left(\boldsymbol{r'}\right)}{\...
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Critical points of the single-layer potential $u(x)=\int_{\partial \Omega}\frac{1}{|x-y|}d\sigma_{y}$ inside a strictly convex bounded domain.
Let $\Omega\subset\mathbb{R}^{3}$ be a bounded strictly convex domain with $C^{2}$ boundary(or smoother). Define $$u(x)=\int_{\partial \Omega}\frac{1}{|x-y|}d\sigma_{y}, \quad x\in \Omega.$$
Question: ...
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Capacity–duty flux split for gated vs fixed window
Let $D\subset\mathbb{R}^3$ be a bounded $C^2$ domain. On its boundary $\partial D$, place two disjoint, tiny patches $S_\varepsilon$ and $T_\delta$ with diameters $\varepsilon,\delta\ll1$.
Consider ...
user1675092
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Does $\Delta u\in \ell^1(\mathbb{Z}^d)$ with bounded first differences force $u$ to be affine?
Let $d\ge 2$. On $\mathbb{Z}^d$ define the discrete Laplacian
$$
\Delta u(x)=\sum_{i=1}^d\big(u(x+e_i)+u(x-e_i)-2u(x)\big),
$$
and for $S\subset\mathbb{Z}^d$ define the upper Banach density
$$
d^*(S)=\...
user1675092
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Poisson's equation in a domain with a hole
Let $N > 2$ and let $\omega, \Omega \subset \mathbb{R}^N$ be open and bounded sets with smooth boundary. Assume both sets contain the origin. For $\sigma > 0$, consider the boundary value ...
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Approximation of (eg, finding accurate bounds for) $\int_{0}^{\sqrt{2}}\frac{\log|r-t|}{(4-t^2)^2}dt$, for $r\in [0, \sqrt{2}]$
I want to understand the following integral $$\int_{0}^{\sqrt{2}}\frac{\log|r-t|}{(4-t^2)^2}dt$$ for values of $r\in [0, \sqrt{2}]$.
What I have tried so far is to used Laurent series expansions, but ...
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Understanding the integral operator $G_\sigma$ defined in Buser's book Geometry and Spectra of Compact Riemann surfaces, Chapter 7
Let $M$ be a closed manifold of dimension $m$. Let $p\in C^{\infty}(M\times M\times (0,\infty); \mathbb R)$ be the heat kernel on $M$. On page 193 of Chapter 7, author defines an integral operator ...
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An integral estimation form Buser's book: Geometry and Spectra of Compact Riemann Surfaces, Chapter 7.
I am reading through the 'Geometry and Spectra of Compact Riemann Surfaces', Chapter 7. I found the following integral estimate, which seemed interesting to me and it took me a while to figure out. ...
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Function with semidefinite Hessian everywhere is convex or concave
Real functions
Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a $\mathcal{C}^\infty$-function, with Hessian matrix $H_f := \left( \frac{\partial^2 f}{\partial x_i \partial x_j} \right)_{i,j = 1}^n$. ...
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Are polar sets lower-dimensional?
For $n\geq2$, $P\subset\mathbb R^n$ is said to be polar if there is some superharmonic function $u$ which is identically $\infty$ on $P$. These sets are generally thought of as small, but I lack ...
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Uniqueness of Single-Layer Potentials and Connection to Dirichlet-to-Neumann Operators
I have a question about some sort of uniqueness in the context of (planar) single-layer potential theory:
Suppose, $ \Omega = \Omega_{\text{in}} \subset \mathbb{R}^2$ is some bounded open subset with (...
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Application of Harnack's inequality
Currently I am reading the Book "Potential Theory in the Complex Plane" by Ransford and I try to solve Exercise 1.3.1. Before I state the exercise, note that $\Delta(0,\rho)$ denotes the ...
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Locally uniform convergence of harmonic functions implies locally uniform convergence of derivatives [duplicate]
Currently I am reading "Potential Theory in the Complex Plane" by Ransford and I try to solve exercise 1.2.3. Which is given by:
Exercise 3: Let $(h_n)_{n\geq1}$ be a sequence of harmonic ...
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Newtonian potential on hyperplane smooth up to boundary?
Let $\rho:\mathbb{R}^2 \to \mathbb{R}$ be smooth and of compact support. We define the newton potential $$u(x,y,z) = \int_{\mathbb{R}^2} \rho(s_1,s_2) \frac{1}{\|(x,y,z) - (s_1,s_2,0) \|} ds_1 ds_2. $$...
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A sort of fundamental calculus theorem
Let $u\in C_c^\infty(\mathbb{R}^n)$. It is known that $$u(x)=\frac{1}{\omega_n}\int Du(y)\cdot \frac{x-y}{|x-y|^n}dy,$$ where $\omega_n$ is the surface are of the unit ball. Other identity is that $$u(...
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Generalised Laplace/Gauss transform of $\frac{\exp(-ax)}{x}$
Let $a>0$ be a constant and suppose $x\in\mathbb R_+$, how to prove：
$\frac{\exp(-ax)}{x}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^\infty \exp(-x^2t^2-\frac{a^2}{4t^2})dt$ ?
This seems like the ...
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Laplacian of 1/r equals zero?
i was reading and suddenly i saw the expression:
$$
\nabla^2 \left(\frac{1}{r}\right) = 0
$$
where $\frac{1}{r} = \frac{1}{\sqrt{x^2 + y^2 + z^2}} $
i do not understand how does it becomes zero with $ ...
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Question on Theorem 2.4 from Chapter VI of Ma and Röckner
In the above mentioned theorem in the book "Introduction to the Theory of (non-symmetric) Dirichlet Forms" the one-to-one correspondence of PCAFs $A$ and smooth measures $\mu$ is expressed ...
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Why does a center manifold appear when a potential flow is perturbed by a constant vector field?
Let $h(\mathbf{x})$ be a solution to the Laplace equation in $\mathbb{R}^2$ or $\mathbb{R}^3$ under some set of boundary conditions. Define $\mathbf{q}(\mathbf{x}) = -\nabla h(\mathbf{x})$. Let $\...
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The lower bound of Green function in small ball
Let $G$ be a Green function on $B_r=B_{r}(0)\subset\mathbb{R}^n$ with $n\geq 3$, namely,
$$
G\left( x,y \right) =\begin{cases}
\Gamma \left( \left| x-y \right| \right) -\Gamma \left( \frac{\left| x \...
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Courant nodal domain theorem for Riesz potential operator
Let $\Omega \subseteq \mathbb{R}^N$ be a bounded open set and $N\geq 2$. Let $0<\alpha<N$.
The Riesz potential operator on $L^2(\Omega)$ is defined by
\begin{equation*}
R_{\alpha,\Omega}(u)(...
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Vector calculus equality from potential theory
Jackson's Classical Electrodynamics, p. 74, does 2.14 -> 2.16 below as integration by parts. I would like to understand the variational approach in Binney & Tremaine's Galactic Dynamics, p.33:
...
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Approximating a given subharmonic function on a Riemann surface, with smooth functions
Let $\Sigma$ be a (connected) Riemann surface and let $u$ be subharmonic on $\Sigma$.
I would like to approximate $u$ from above with smooth subharmonic functions, in the following sense:
For any $U\...
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How to prove the positivity of the first eigenfunction of Logarithmic potential operator on $L^2(\Omega)$?
I am considering the operator $$\mathcal{L} u(x)= \frac{1}{2\pi}\int_\Omega \log\dfrac{1}{|x-y|}u(y)dy\ \ \text{on} \ \ L^2(\Omega)$$ where $\Omega$ is a bounded domain such that $\mathcal{L}_\Omega$ ...
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Are this kind of plurisubharmonic function continuous?
Let $E$ be a closed pluripolar set in $\Omega \in \mathbb{C}^n$, if $u$ is a plurisubharmonic function on $\Omega$ and continuous on $\Omega$\E, is $e^u$ a continuous function on $\Omega$?
I think the ...
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Is there some link between analytic continuation and potential theory?
It seems like the analytic continuation of a function has a lot in common with the process of trying to define a potential for a vector field (or a differential 1-form). In particular, an analytic ...
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Inconsistency in the Fundamental Solution Constant for the Laplacian in Higher Dimensions
I am working through a problem involving the extension problem for fractional Laplacians, and I've encountered some inconsistencies in the derivation of the fundamental solution and the associated ...
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Complex Potential Between Axes and a Hyperbola
I'm looking at the exact same question as here:
Complex potential between axes & hyperbola
(Advanced Engineering Mathematics 8th edition, Erwin Kreyszig, problem 12 of section 16.1)
Find the ...
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Why the log function is so important on the plane?
I am studying right now some Complex Analysis and I have seen the importance of the (complex) logarithm function in almost every subject in it. Now I'm intrigued with that (possible) relation between $...
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An inequality for continuity of normal derivative of double layer potential
Let $\Gamma \subset \mathbb{R}^2$ be a $C^2$ smooth Jordan curve, $t_x$,$n_x$ be the tangent and exterior normal of $x \in \Gamma$ respectively.
$$
U(x,t_x,\delta,n_x,\delta) := \{ y = x + \xi t_x + \...
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Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$
Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following:
Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
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Is there a Dirichlet form on product space?
Let $(X_t)_{t\ge 0}$ be a Markov process on a locally compact Polish space $E$ (say for example the Sierpinski Gasket). Then there is a Dirichlet form associated with $X_t$ on $E$, call it $({\cal E},{...
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Solving the Poisson equation
I am an undergraduate student, in this semester I am taking the course of partial differential equations.
So reading about Poisson equation by Evan's classic book for pdes, i have some questions:
...
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Potential density of killed brownian by local time
Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(-...
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How to calculate the derivative of a Newtonian potential inside a box with uniform source distribution?
I'm working on a potential flow problem. I have a box, centered at the origin (i.e. $V=[-x_b,x_b]\times[-y_b,y_b]\times[-z_b,z_b]$) that has inside of it a uniform distribution of source strength. We ...
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Chetaev theorem for discrete time
In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations
Theorem 1 seems to prove a discrete-time analog of Chetaev ...
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Indeterminate Form for Partial Derivative of Flow Variables
The following is a paraphrased version of the derivation within John D. Anderson's Fundamentals of Aerodynamics's section on the Method of Characteristics:
The exact governing equation for two-...
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Electrostatic potential due to repeating coplanar charged strips (Mathews&Walker 5.1)
This problem is due to Mathews and Walker's Mathematical Methods of Physics, exercise 5.1.
On the 2D plane, suppose we have a series of coplanar charged strips of line charge density $\lambda$ and ...
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Riesz potential cannot be extended to $L^1 \rightarrow L^{\frac{n}{n-s}}$ bounded operator
Riesz potential is defined as
$$I_s(f) = K_s*f$$
for Schwartz function $f$ with $K_s(x) = c_s|x|^{-n+s}$.
By weak Young's inequality, $I_s$ can be extended to $L^p \rightarrow L^q$ bounded operator ...
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Different definitions of conformal capacity
A condenser is a pair $(D,E)$, where $D$ is a domain in the plane and $E$ is a compact subset of $D$. The capacity of the condenser $(D,E)$ is defined by:
$$\text{cap}(D,E) = \inf \int_{D} |\nabla u|^...
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How to evaluate a volume integral with a singular interior point?
I would like to evaluate the following integral:
$I = \int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} \frac{x-x_0}{[(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2]^{3/2}} dz dy dx$
where the fixed point $(x_0,...
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Level set of green function
$\textbf{Background for problem statement}$:
Let $B \subset \mathbb{C}$ be a bounded domain, and $g_{B}(z,z_0) = g$ its Green's function with pole at $z_0 \in B$, so $g$ is harmonic in $B \setminus \{...
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Integral for the surface area of a half n-sphere
I am trying to evaluate the following integral on $\mathbb{R}^{n-1}$
$$\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n}{2}}}dx$$
I claim that this is equal to the half the surface area of the ...
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Green function for the upper-half space $\mathbb{R}^{n+1}_+$
Given $\mathbb{R}^{n+1}_+=\{X=(x,t): x \in \mathbb{R}^n , t>0 \}$ as domain, what is the explicit formula for the Green function $G(X,Y)$ for the Laplacian on $\mathbb{R}^{n+1}_+$? I know that ...
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proof related to jump relation in double-layer potential
Assume $D \subset \mathbb{R}^{n}$ be a bounded and open domain with $C^{2}$ boundary. Let $x \in \partial D$ and $r>0$. Define
\begin{align*}
C_{r} := D \cap \partial B( x,r) \text{ and } \...
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Can we construct a compact complete pluripolar set?
We know that analytic sets are not relatively compact in $\mathbb{C}^n$ when $n\geq 2$. In fact, we can construct plurisubharmonic functions by holomorphic functions. So analytic subset belong to ...
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Prove $ \lim_{\varepsilon\to 0} \frac{dH^{n-1}(\Omega\cap\partial B(x,\varepsilon))}{dH^{n-1}(\partial B(x,\varepsilon))} = \frac{1}{2}.$
Assume $\Omega\subset \mathbb{R}^n$ is a bounded open set with Lipschitz boundary; let $d H^{n-1}$ be the Hausdorff measure on $\partial \Omega$.
Since $\Omega$ has Lipschitz boundary, for $d H^{n-1}$-...
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