Questions tagged [potential-theory]

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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 ...
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0 answers
62 views

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)...
Ryan Li's user avatar
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1 answer
126 views

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) = - \...
jcb2535's user avatar
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1 vote
0 answers
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 ...
Joseph Van Name's user avatar
2 votes
1 answer
135 views

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 $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
Focus's user avatar
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0 answers
30 views

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-...
Sarvesh Ravichandran Iyer's user avatar
3 votes
0 answers
298 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 ...
Mingchen Xia's user avatar
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 $\mathbb{B}^n$ ...
Piero D'Ancona's user avatar
3 votes
1 answer
121 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 ...
Piero D'Ancona's user avatar
2 votes
1 answer
295 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 ...
Oscar Sucre's user avatar
1 vote
2 answers
220 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 ...
asv's user avatar
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2 votes
1 answer
208 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 $C^2$-...
asv's user avatar
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2 votes
1 answer
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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 ...
Analyse300's user avatar
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0 answers
40 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 ...
Matt Rosenzweig's user avatar
3 votes
0 answers
105 views

L¹ norm of Riesz potentials on flat tori

Let $g$ be the distribution whose Fourier coefficients are given by $$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$ ...
Matt Rosenzweig's user avatar
2 votes
0 answers
95 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 ...
anon's user avatar
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1 answer
83 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 ...
Analyse300's user avatar
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0 answers
71 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 ...
xin fu's user avatar
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4 votes
1 answer
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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 ...
Analyse300's user avatar
2 votes
0 answers
119 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")...
naruto's user avatar
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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-...
5th decile's user avatar
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1 vote
0 answers
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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 ...
user91126's user avatar
  • 213
5 votes
0 answers
155 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 ...
maxematician's user avatar
3 votes
0 answers
173 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 ...
asv's user avatar
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1 vote
1 answer
131 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?
asv's user avatar
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1 vote
0 answers
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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; ...
ABIM's user avatar
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6 votes
0 answers
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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)...
ABIM's user avatar
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1 vote
0 answers
55 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 ...
Dustin G. Mixon's user avatar
3 votes
1 answer
271 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 ...
T. Huynh's user avatar
3 votes
0 answers
118 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 | ...
Kosh's user avatar
  • 356
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 -...
asv's user avatar
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0 votes
1 answer
386 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\...
asv's user avatar
  • 21.1k
6 votes
2 answers
820 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 ...
S. Euler's user avatar
  • 285
5 votes
0 answers
178 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 \...
Luca Benatti's user avatar
2 votes
1 answer
151 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 ...
asv's user avatar
  • 21.1k
0 votes
0 answers
132 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.
M. Rahmat's user avatar
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0 answers
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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 ...
M. Rahmat's user avatar
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2 votes
0 answers
66 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 $...
M. Rahmat's user avatar
  • 411
3 votes
1 answer
264 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 ...
Ali Taghavi's user avatar
1 vote
0 answers
64 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 ...
M. Rahmat's user avatar
  • 411
3 votes
0 answers
185 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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
sharpe's user avatar
  • 701
2 votes
0 answers
113 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})...
GaC's user avatar
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1 vote
0 answers
79 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 ...
M. Rahmat's user avatar
  • 411
0 votes
2 answers
167 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 $D$ ...
M. Rahmat's user avatar
  • 411
1 vote
0 answers
60 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 $...
M. Rahmat's user avatar
  • 411
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(...
sharpe's user avatar
  • 701
1 vote
1 answer
51 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 ...
M. Rahmat's user avatar
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0 votes
0 answers
131 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 $...
sharpe's user avatar
  • 701
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 ...
M. Rahmat's user avatar
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0 answers
44 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(...
M. Rahmat's user avatar
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