Questions tagged [potential-theory]
The potential-theory tag has no usage guidance.
203
questions
27
votes
1
answer
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views
A problem of potential theory arising in biology
Let $K_0$ and $K_1$ be two bounded, disjoint convex sets in $R^n,n\geq 3$,
and
$u$ the equibrium potential, that is the
harmonic functon in $R^n\backslash\{ K_0\cup K_1\}$ such that $u$
has boundary ...
- 88.9k
25
votes
1
answer
2k
views
The origin of Discrete `Liouville's theorem'
It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...
- 11.9k
15
votes
0
answers
386
views
References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
14
votes
2
answers
2k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,095
10
votes
1
answer
747
views
Canonical English edition of Dellacherie and Meyer's "Probabilities and Potential"
Probabilities and Potential by Dellacherie and Meyer is a "bible" of probabilistic potential theory, Markov processes, and many related topics. I want my library to acquire it, but I am a bit ...
- 29.2k
10
votes
1
answer
423
views
Modern version of an inequality of R. M. Gabriel for contour integrals
I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...
- 6,186
10
votes
0
answers
307
views
Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality
$\newcommand\rank[1]{\lvert#1\rvert}$Let $\Bbb{P}$ be a 1-differential poset with a unique bottom element $\emptyset \in \Bbb{P}$. With some minor abuse in terminology, The
Plancherel measure state $...
- 1,847
9
votes
4
answers
2k
views
Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
- 2,102
9
votes
2
answers
410
views
Core for a Sobolev space
Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. The first-order Sobolev space $W^{1,2}(D)$ on $D$ is defined by
\begin{align*}
W^{1,2}(D)=\{f \in L^2(...
- 701
9
votes
1
answer
282
views
Which domain maximizes the energy of the Lebesgue measure?
This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...
- 13.7k
8
votes
2
answers
7k
views
Version of the Poincaré Inequality
Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|_p \le C \|\nabla u\|_p$$
($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
- 2,210
8
votes
1
answer
654
views
How does electric potential relate to mean curvature?
Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.
Now consider the electric potential generated by this uniform mass ...
- 2,601
8
votes
0
answers
130
views
Functions between Markov chains that preserve local harmonicity
Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...
- 19.4k
7
votes
2
answers
1k
views
Boundary regularity for the Dirichlet problem
Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator.
We wish to solve the Dirichlet problem (...
- 373
7
votes
0
answers
237
views
Sard's theorem for superharmonic functions: less regularity required?
A function $f:\mathbb{R}^d \to \mathbb{R}$ must be at least $C^d$ in order to guarantee in general that
$$\{\phi\in \mathbb{R}|\,\exists x\in \mathbb{R}^d:\,f(x)=\phi,\,(\nabla f)(x)=0\}$$
is a zero-...
- 1,451
7
votes
0
answers
200
views
Minimizing energy on $\mathbb{S}^2$ for absolutely monotonic type potentials
For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a ...
- 3,177
7
votes
0
answers
491
views
planar mappings that preserve elliptic measure
Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...
- 551
6
votes
2
answers
822
views
$\log |f|$ is subharmonic
It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions:
(1) Are there some weaker ...
- 285
6
votes
1
answer
361
views
electron configuration on manifolds
Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...
- 543
6
votes
2
answers
2k
views
Logarithmic potential
Given a continuous, compactly supported function $f$ on $R^2$, it is known that the logarithmic potential of $f$, that is
$$
U_{f}(x):=-\frac{1}{2\pi}\int\log|x-y|f(y)dy
$$
has the following decay at ...
- 61
6
votes
1
answer
633
views
Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$
The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...
- 5,163
6
votes
1
answer
477
views
A finely open set, not open up to polar set?
I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
- 1,474
6
votes
0
answers
181
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 4,969
6
votes
0
answers
372
views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
- 61
5
votes
3
answers
1k
views
Calculation of logarithmic capacity?
I am reading this paper about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like ...
- 1,019
5
votes
2
answers
397
views
Most general conditions for (weak or classical) solutions to Poisson's equation
I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving
$$
\left\{
\begin{align}
& \Delta \psi = - \rho & & \mbox{in } \mathbb{...
- 401
5
votes
1
answer
382
views
Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle
What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The answer appears to be uniform measure, since informally it appears better to have ...
- 3,177
5
votes
2
answers
1k
views
Kähler potentials that depend only on geodesic distance
Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are ...
- 1,368
5
votes
1
answer
279
views
Newtonian potentials of balls and spheres
This is a simple question whose answer was probably known to Poisson, but I was not able to find it by searching. I need explicit formulas for the Newtonian potential of the unit ball $\mathbb{B}^n$ ...
- 8,775
5
votes
1
answer
316
views
Convergence in energy of bounded (semi)subharmonic functions
Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...
- 650
5
votes
1
answer
150
views
Superharmonicity at infinity
Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing ...
- 411
5
votes
1
answer
201
views
How to choose contour for rational approximation
Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\...
- 243
5
votes
0
answers
156
views
Potential theory as a tool in extrinsic flows
Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
- 197
5
votes
0
answers
179
views
$p$-capacity of the closure
The $p$-capacity of a condenser $(K,\Omega)$ with $K$ compact and $\Omega$ open bounded is defined as
$$
\mathrm{Cap}_p(K,\Omega)=\inf \left\lbrace \int_{\Omega} |\nabla u|^p \mathrm{d} x : u \in \...
- 151
5
votes
0
answers
94
views
Stability and capacity, error in the book of Adams-Hedberg?
I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...
- 93
4
votes
3
answers
476
views
Positivity of the Coulomb energy in two dimensions
In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...
- 255
4
votes
2
answers
279
views
An extremal type problem on segments
I am interested in the following extremal-type problem.
Let us define $\Psi$ by
$$\Psi(x)=\max_{f\in L^2[0,x] \,\,\text{with}\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$
on $(0,\...
- 41
4
votes
1
answer
68
views
Singularities on null capacity sets are removable — Wiener or Bouligand?
A classical theorem on harmonic functions states that singularities of bounded harmonic functions are removable if the singular set is of null capacity. This theorem is sometimes attributed to ...
- 3,292
4
votes
1
answer
137
views
Boundedness of Riesz potential on Hardy space
I encounter the following claim in one paper:
If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$,...
- 593
4
votes
1
answer
703
views
Elliptic, parabolic and hyperbolic Riemann surfaces: classification?
In the book of Kra and Farkas on Riemann surfaces the following (slightly unusual) definition is given:
Definition IV.3.2 (Section IV.3). Let $M$ be a Riemann surface. We will call $M$ elliptic if and ...
- 14k
4
votes
1
answer
266
views
Limit for series of Bessel functions evaluated at zeros
The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}
$$
where $J_i$ is the Bessel function of $i^{th}$ ...
- 573
4
votes
1
answer
445
views
multivalued holomorphic function on Riemann surfaces
Let $M$ be an open Riemann surface and $f$ a multivalued holomorphic function from $M$ to $\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. Suppose that the monodromy of $f$ lies in the two-...
- 371
4
votes
2
answers
162
views
What fraction of a charge is induced on a surface via balayage?
Consider a smooth, bounded domain $\Omega \subset \mathbb{R}^3$, and place a charge $q>0$ at a point $z\in\mathbb{R}^3\setminus\overline\Omega$. Via the concept of balayage, there is an induced ...
- 401
4
votes
2
answers
310
views
Boundary Value Problem in the space of Distributions
I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...
- 41
4
votes
1
answer
222
views
Show those PSH functions belongs to Sobolev space
Let u be a plurisubharmonic function defined on the unit ball $\mathbb{B}$ of $\mathbb{C}^{k}$ such that $u \ge 1$.
Question: why the partial derivates $\frac{\partial u}{\partial x_{i}}$ (which are ...
- 83
4
votes
1
answer
168
views
Restricted Perron-Bremermann envelopes
Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization
$$
\phi^*(z) = \...
- 151
4
votes
1
answer
357
views
reference request: Riesz/Newton potential and HLS inequality in L1.logL1
Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$
$$
f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
- 5,163
4
votes
0
answers
177
views
Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice
This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $\Bbb{YF}$, namely:
Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality
...
- 1,847
4
votes
0
answers
130
views
regularity of zero point
We consider 1-d process $X$
$$ X(t) = b t + J_{t} + M_{t}$$
where $b$ is constant, $M$ is a continuous martingale process with
$M(0) = 0$, and
$J$ is a symmestric $\alpha$-stable process with its ...
- 1,369
3
votes
1
answer
195
views
Capacity of a unit disk with a small bump
Let $A_r = \{z\in\mathbb{C}: |z|\leq 1\}\cup\{z\in\mathbb{C}: |z-1|\leq r\}$ be the unit disk with a small "bump" (I'm interested in the regime $r\to 0$). What can be said about the logarithmic ...
- 1,897