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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. ...
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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 ...
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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 ...
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Let $U \subset\mathbb{R}$$n$ be an arbitrary connected bounded open set, and let $f$ : $∂U$ $\rightarrow$ $\mathbb{R}$ be continuous, where $∂U$ denotes the set $\overline{U} - U$. Are there ...
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In a paper that I am reading, the following equality is stated ($s,p>0$ and $|S^{n−1}|$ the measure of the $(n−1)$-dimensional sphere) $$ \left(s \int_{\mathbf{R}^n} \int_{|y| \geqslant|x|} \frac{d ...
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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 ...
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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 ...
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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 ...
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From potential theory, I know the following double integral must be strictly positive. $$\int_{\mathbb{S}^2} \int_{\mathbb{S}^2} \frac{\eta_3^3 \xi_3^3}{|\xi - \eta|} \, \mathrm{d}\eta \, \mathrm{d}\...
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Let $s\in \mathbb{C}$ with $0\leq \operatorname{Re}s<\infty$, $f\in \mathcal{S}'(\mathbb{R}^n)$ and $\xi\in \mathbb{R}^n$. We define the Bessel potential $\Lambda_s$ of order $s$ of $f$ as $$\...
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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 - ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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)...
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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) = - \...
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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 ...
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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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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 ...
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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 $\mathbb{B}^n$ ...
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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 ...
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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 ...
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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 ...
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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 $C^2$-...
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$\DeclareMathOperator\PSH{PSH}$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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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 ...
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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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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 ...
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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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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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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
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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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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-...
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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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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
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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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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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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; ...
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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 $$ \int_{x\in X}\,d(y_0,f(x)...
AB_IM's user avatar
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Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that $$ \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, $$ with ...
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Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
T. Huynh's user avatar
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Let $\Omega \subseteq \mathbb{R}^n$ be a measurable set of finite measure. It is well-known that there holds $$ \sup_{x \in \mathbb{R}^n} \int_{\Omega} \frac{d z}{| x - z |^{n - 1}} \leqslant c_n | ...
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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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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
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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
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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 \...
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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 ...
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