Questions tagged [potential-theory]
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218 questions
2 votes
0 answers
139 views
Possible flaw in Karp and Margulis's proof of convexity of null quadrature domains?
I was reading L. Karp and A. Margulis's proof of the convexity of the complement of a null quadrature domain. This paper is cited by many others, for example, S. Eberle, A. Figalli and G. Weiss. ...
- 189
3 votes
1 answer
475 views
Reference of a maximum principle used in a paper written by Brezis and Merle
In the paper "Uniform estimates and Blow-up behavior for solutions of $-\Delta u =V(x)e^u$ in two dimensions" in the Theorem 1 (A basic inequality), we have the following result: Let $\Omega ...
- 305
1 vote
1 answer
300 views
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 ...
3 votes
1 answer
403 views
Existence of a harmonic function on a bounded open set in euclidean space extending a continuous function on its boundary
Let $U \subset\mathbb{R}$$n$ be an arbitrary connected bounded open set, and let $f$ : $∂U$ $\rightarrow$ $\mathbb{R}$ be continuous, where $∂U$ denotes the set $\overline{U} - U$. Are there ...
- 3,555
0 votes
1 answer
137 views
Potential theoretic integral
In a paper that I am reading, the following equality is stated ($s,p>0$ and $|S^{n−1}|$ the measure of the $(n−1)$-dimensional sphere) $$ \left(s \int_{\mathbf{R}^n} \int_{|y| \geqslant|x|} \frac{d ...
- 410
2 votes
0 answers
138 views
Name for integral representation of Riesz potential
Let $\phi:\mathbb{R}^d \rightarrow \mathbb{R}$ be a sufficiently nice (e.g. Schwartz) radial function . Then it is classical by scaling that the Riesz potential $|x|^{-s}$, for $s>0$, may be ...
- 903
3 votes
1 answer
217 views
The computation of certain integration on Euclidean sphere
Consider the Euclidean unit open ball $B^3(0)\subset \mathbb{R}^3$ centered at the origin with boundary $S^2$. Fix a point $y\in \mathbb{R}^3$ with $|y|>1$. For $x\in B^3(0)$, define the function ...
- 127
1 vote
0 answers
109 views
Potential-theoretic estimate for heat kernel convolution appeared in J. Serrin's paper
I'm reading James Serrin's paper https://doi.org/10.1007/BF00253344 and stucked at page 193. After proving $u\in L_t^{\infty}L_x^{\infty}$ and $\omega\in L_t^{\infty}L_x^{\infty}$, he then wants to ...
- 113
1 vote
1 answer
138 views
Positivity of double integral II
From potential theory, I know the following double integral must be strictly positive. $$\int_{\mathbb{S}^2} \int_{\mathbb{S}^2} \frac{\eta_3^3 \xi_3^3}{|\xi - \eta|} \, \mathrm{d}\eta \, \mathrm{d}\...
- 410
2 votes
1 answer
161 views
Boundness of purely imaginary powers of Bessel potential
Let $s\in \mathbb{C}$ with $0\leq \operatorname{Re}s<\infty$, $f\in \mathcal{S}'(\mathbb{R}^n)$ and $\xi\in \mathbb{R}^n$. We define the Bessel potential $\Lambda_s$ of order $s$ of $f$ as $$\...
1 vote
1 answer
151 views
Positivity of caloric measure density on a cylinder
Let $u$ be a solution to the heat equation $u_t = \Delta u$ in the unit cylinder $B_1\times(-1,0) \subset \mathbb R^{n+1}$. Then, it is well known (see for instance Chapter 2 in "Watson - ...
4 votes
0 answers
268 views
What is the maximum tidal force between two objects with unit volumes and unit density?
Motivation for this problem This problem arises from the fact that the derivative of the gravitational force (tidal force) in the $z$-direction between two objects $A$ and $B$, which have equal ...
- 273
3 votes
0 answers
252 views
Green function of an elliptic operator
Let $L$ be an elliptic operator on $\Bbb C$ with $$\DeclareMathOperator{\Img}{Im} L^{-1}f(z)=\int_{\Bbb {C}}K_1(|z-w|^2)f(w) e^{i\cdot\Img\langle z,\overline{w}\rangle} \, dw $$ where $K_1$ is the ...
- 109
2 votes
0 answers
132 views
Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator. My question is there is a formula of the ...
- 21
1 vote
0 answers
367 views
Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
2 votes
0 answers
178 views
Estimating an integral of the Green function in the plane
Suppose $\Omega$ is a bounded, simply connected domain, $z_{0}\in{\Omega}$ and for any $z\in{\Omega}$, $d_{z}:=\text{dist}(z,\partial{\Omega})$. I am interested in understanding the behavior of ...
- 672
4 votes
1 answer
223 views
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 ...
- 470
2 votes
0 answers
100 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)...
- 31
1 vote
1 answer
301 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) = - \...
- 67
1 vote
0 answers
58 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 ...
- 29k
2 votes
1 answer
238 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 $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
- 177
3 votes
0 answers
397 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
413 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 $\mathbb{B}^n$ ...
- 9,142
3 votes
1 answer
284 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 ...
- 9,142
2 votes
1 answer
607 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
488 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 ...
- 23k
2 votes
1 answer
345 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 $C^2$-...
- 23k
2 votes
1 answer
264 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 ...
- 83
1 vote
0 answers
73 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 ...
- 903
3 votes
0 answers
166 views
L¹ norm of Riesz potentials on flat tori
Let $g$ be the distribution whose Fourier coefficients are given by $$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$ ...
- 903
2 votes
0 answers
124 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
121 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 ...
- 83
0 votes
0 answers
102 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 ...
- 633
4 votes
1 answer
282 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
2 votes
0 answers
158 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
7 votes
0 answers
263 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-...
- 1,461
1 vote
0 answers
116 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
182 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 ...
- 237
3 votes
0 answers
219 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 ...
- 23k
1 vote
1 answer
176 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?
- 23k
1 vote
0 answers
77 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; ...
- 4,942
6 votes
0 answers
215 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 $$ \int_{x\in X}\,d(y_0,f(x)...
- 4,942
1 vote
0 answers
77 views
Are sharper lower bounds known for these potentials on the sphere?
Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that $$ \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, $$ with ...
- 8,095
3 votes
1 answer
341 views
Definition of Martin kernels
Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
- 41
3 votes
0 answers
229 views
Riesz potential on the boundary of a smooth domain
Let $\Omega \subseteq \mathbb{R}^n$ be a measurable set of finite measure. It is well-known that there holds $$ \sup_{x \in \mathbb{R}^n} \int_{\Omega} \frac{d z}{| x - z |^{n - 1}} \leqslant c_n | ...
- 410
2 votes
0 answers
111 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 -...
- 23k
0 votes
1 answer
598 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\...
- 23k
6 votes
2 answers
1k 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
5 votes
0 answers
205 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
197 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 ...
- 23k