# Questions tagged [potential-theory]

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203
questions

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### Relations between two definitions of harmonic measure

I came into two definitions of harmonic measure on a Riemann surface. The first is defined on p.180 of Riemann surfaces, 2nd by Kra and Farkas, which read as follows.
Theorem. Let $M$ be a hyperbolic ...

2
votes

0
answers

62
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### Localized estimate for divergence free vector field

Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...

0
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1
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126
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### Green's function in terms of logarithmic potential and energy of a measure

Let $\mu$ be a finite (Borel) measure on $\mathbb{C}$ with compact support $K := \mbox{supp } \mu$.
The logarithmic potential associated to the measure $\mu$ is
\begin{equation}
\Phi_{\mu}(z) = - \...

1
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0
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42
views

### Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices

Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...

2
votes

1
answer

135
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### Is every simply connected domain regular?

Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$...

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### Concerning the conversion of an essential supremum to a pointwise estimate

In the following paper :
Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian, Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3747-...

3
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298
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### 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 ...

5
votes

1
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279
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### 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 $\mathbb{B}^n$ ...

3
votes

1
answer

121
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### 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 ...

2
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1
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295
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### 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 ...

1
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2
answers

220
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### 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 ...

2
votes

1
answer

208
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### 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 $C^2$-...

2
votes

1
answer

154
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### 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 ...

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0
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40
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### 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 ...

3
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105
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### L¹ norm of Riesz potentials on flat tori

Let $g$ be the distribution whose Fourier coefficients are given by
$$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$
...

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95
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### 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 ...

0
votes

1
answer

83
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### 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 ...

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71
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### 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 ...

4
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1
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222
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### 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 ...

2
votes

0
answers

119
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### 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")...

7
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237
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### 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
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0
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85
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### 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 ...

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155
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### 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 ...

3
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173
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### 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 ...

1
vote

1
answer

131
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### 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?

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0
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52
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### 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; ...

6
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181
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### 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)...

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0
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55
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### 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 ...

3
votes

1
answer

271
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### 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 ...

3
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0
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118
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### 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 | ...

2
votes

0
answers

85
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 -...

0
votes

1
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386
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### 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\...

6
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2
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820
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### $\log |f|$ 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 ...

5
votes

0
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178
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### $p$-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 \...

2
votes

1
answer

151
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### 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 ...

0
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0
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132
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### 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.

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### 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 ...

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### 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 $...

3
votes

1
answer

264
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### 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 ...

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0
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64
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### 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 ...

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### 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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...

2
votes

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113
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### 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})...

1
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### 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 ...

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2
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167
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### 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 $D$ ...

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0
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60
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### 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 $...

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2
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410
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### 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(...

1
vote

1
answer

51
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### 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 ...

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131
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### 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 $...

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150
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### 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 ...

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### 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(...