Questions tagged [potential-theory]
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196
questions
3
votes
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views
Demailly regularisation on singular complex spaces
Let $X$ be a compact (Hausdorff reduced) complex space. It is asserted (and used in an essential way) in a famous paper by Demailly and Păun ("Numerical characterization of the Kähler cone of a ...
- 329
5
votes
1
answer
227
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 $\mathbb{B}^n$ ...
- 8,446
3
votes
1
answer
85
views
Subharmonic distributions on the plane
A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution ...
- 8,446
2
votes
1
answer
163
views
Value of $\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}$
I wonder if any of you knows how to find the value
of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$
This function shows up while solving a magnetostatic problem
with complex-valued ...
- 29
1
vote
2
answers
135
views
A characterization of plurisubharmonic functions
Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
Recall that $u$ is called plurisubharmonic (psh) if its restriction ...
- 20.3k
2
votes
1
answer
135
views
A possible characterization of subharmonic functions
Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$-...
- 20.3k
2
votes
1
answer
115
views
Proof of a theorem in degenerate Monge Ampère equation by Vincent Guedj and Ahmed Zeriahi
$\DeclareMathOperator\PSH{PSH}$This question is about Proposition 9.25 page 252 from the book "Degenerate Complex Monge-Ampère Equations" by Vincent Guedj and Ahmed Zeriahi (see picture ...
- 63
0
votes
0
answers
29
views
Functional inequality for fractional Laplacian
Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
- 923
3
votes
0
answers
90
views
L¹ norm of Riesz potentials on flat tori
Let $g$ be the distribution whose Fourier coefficients are given by
$$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$
...
- 923
2
votes
0
answers
83
views
What does a Lipschitz barrier imply about boundary regularity of a domain?
Consider the Dirichlet problem for Laplace's equation in a bounded domain $\Omega \subset \mathbb R^n$:
$$
-\Delta u = 0, \quad x \in \Omega,
$$
with $u = \phi$ on $\partial\Omega$, and $\phi$ is ...
- 21
0
votes
1
answer
80
views
Convergence of Riesz measure of SH function
Let $u$ be a subharmonic function in a domain $\Omega$ pf $\mathbb{C}$. The functions $u_{j} := \max(u, -j)$ still subharmonic. Let $\mu := \Delta u$ and $\mu_{j} := \Delta u_{j}$ be the associated ...
- 63
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votes
0
answers
65
views
When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?
Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...
- 603
4
votes
1
answer
199
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 ...
- 63
2
votes
0
answers
102
views
On the definition of Cauchy transform [closed]
I have seen two different definitions of the Cauchy transform of a smooth function one is with respect to the line integral (for eg. in. the book "The Cauchy transform and potential theory")...
- 21
6
votes
0
answers
171
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-...
- 1,369
1
vote
0
answers
77
views
Target space of Green's operator on $L^p$-differential forms on closed manifolds
Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
- 213
5
votes
0
answers
149
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 ...
- 143
3
votes
0
answers
153
views
A question on the proof of Bedford-Taylor theorem in Demailly's book
I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions.
I am reading a proof in the ...
- 20.3k
1
vote
1
answer
122
views
Equality of two subharmonic functions
Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset.
Is it true that $u=v$ everywhere?
- 20.3k
1
vote
0
answers
44
views
Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
- 4,881
6
votes
0
answers
172
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
$$
\int_{x\in X}\,d(y_0,f(x)...
- 4,881
1
vote
0
answers
53
views
Are sharper lower bounds known for these potentials on the sphere?
Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2,
$$
with ...
- 7,325
2
votes
1
answer
194
views
Definition of Martin kernels
Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
- 31
3
votes
0
answers
99
views
Riesz potential on the boundary of a smooth domain
Let $\Omega \subseteq \mathbb{R}^n$ be a measurable set of finite measure. It
is well-known that there holds
$$ \sup_{x \in \mathbb{R}^n} \int_{\Omega} \frac{d z}{| x - z |^{n - 1}}
\leqslant c_n | ...
- 356
2
votes
0
answers
60
views
Second order estimates for Dirichlet problem for complex Monge-Ampere equation
Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
- 20.3k
0
votes
1
answer
237
views
Harmonic functions in infinite domain in Euclidean space
EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
- 20.3k
5
votes
2
answers
604
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 ...
- 275
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votes
0
answers
165
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
2
votes
1
answer
142
views
Comparing integrals of bounded subharmonic functions
Let $\Omega \subset \mathbb{R}^n$ be an open open subset. Let $u,v\colon \Omega\to \mathbb{R}$ be two functions such that at least one of them is compactly supported. Assume each of $u$ and $v$ can be ...
- 20.3k
0
votes
0
answers
123
views
Harmonic measure of a punctured disc
Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.
- 411
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votes
0
answers
83
views
Extension of super harmonic functions
The following is stated as an exercise in the "Classical Potential Theory" of Armitage and Gardiner (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open set ...
- 411
2
votes
0
answers
61
views
A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
- 411
3
votes
1
answer
240
views
Harmonic interpolation with analytic initial condition
Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold.
Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function.
Is there a Harmonic ...
- 139
1
vote
0
answers
60
views
Extension of the Kelvin transform
Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
- 411
3
votes
0
answers
174
views
Regularity of harmonic functions for a degenerate elliptic operator
This is a question on a degenerate elliptic operator.
Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
- 609
2
votes
0
answers
107
views
Divergence-free constraint for a boundary integral equation
Consider the system
$$
\begin{cases}
\operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\
c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
- 163
1
vote
0
answers
69
views
Superharmonicity of the distance function
Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
- 411
0
votes
2
answers
150
views
A question on minimum principle
Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...
- 411
1
vote
0
answers
58
views
An open set whose complement is non-thin at infinity
Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that
$$x^*=|x|^{-2}x.$$
For a set $E$, we set $...
- 411
9
votes
2
answers
377
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(...
- 609
1
vote
1
answer
49
views
Modifying a superharmonic function on a neighborhood of infinity
Let $u(x)=\alpha+\beta U(x)$, where $U(x)=|x|^{2-m}$ ($N\geq3$) is the fondamental harmonic function, $\alpha<0$ and $\beta>0$. We know that $u$ is superharmonic on $\mathbb{R}^m$ and harmonic ...
- 411
0
votes
0
answers
109
views
Diffusion process, hitting times, and harmonic functions
Let $E$ be a locally compact metric space. We consider a diffusion process $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in E})$ on $E$ whose lifetime $\zeta$ may be finite: $P_x(\zeta<\infty)>0$ for some $...
- 609
5
votes
1
answer
139
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
0
votes
0
answers
43
views
Superharmonic extension 3
This question is related to the MO post
Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$,
$$u(...
- 411
1
vote
1
answer
162
views
Superharmonic extension 2
This question is a simplified version of the one in the MO post Superharmonic extension.
Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
- 411
4
votes
0
answers
159
views
Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice
This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $\Bbb{YF}$, namely:
Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality
...
- 1,757
0
votes
0
answers
150
views
Inverse of the Riesz potential of a measure
Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$,
$$
I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy.
$$
Assuming $f$ ...
- 3,596
0
votes
0
answers
154
views
Superharmonic extension
We know the following classic result. If $K\subset\mathbb{R}^m$ ($m>1$) is a compact set and $u$ is superharmonic on a neighborhood of $K$, then we can extend $u$ to a superharmonic function $\...
- 411
4
votes
0
answers
125
views
Restricted Perron-Bremermann envelopes
Consider an upper semicontinuous function $\phi: \Omega \to (-\infty, \infty]$, in the sense that $\phi = \phi^*$, where $\phi^*$ denotes the upper semicontinuous regularization
$$
\phi^*(z) = \...
- 141
4
votes
1
answer
64
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 ...
- 3,282