Questions tagged [potential-theory]
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10 questions from the last 365 days
1
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1
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74
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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
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0
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235
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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
0
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0
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72
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Regularity estimates of Double Layer potential
Let $\Omega$ is a bounded open subset of $\mathbb{R}^n,n\ge 2$ with $C^{\infty}$ boundary. Define $$I\left[ \phi \right](x) := -\frac{1}{\omega_n}\int_{\partial \Omega} \frac{(x-y)\cdot \nu_y}{|x-y|^n}...
- 69
3
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0
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179
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Green function of an elliptic operator
Let $L$ be an elliptic operator on $\Bbb C$ with
$$\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
$$ where $K_1$ is the ...
- 113
2
votes
0
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85
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Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator.
My question is there is a formula of the ...
- 21
1
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0
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338
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Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
0
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0
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34
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No jump of hypersingular integral near boundary under lower regularity
Let $\Gamma \subset \mathbb{R}^2 $ be a $C^2 $ smooth simple closed curve, the elasticity double layer potential on $\Gamma $ is defined as
$$
(Wu)(x):= \int_\Gamma (T(\partial_y,n(y))E(x,y))^T u(y) ...
- 269
0
votes
0
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19
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discontinuity of double layer potential kernel on $\mathbb{R}^2 \times \Gamma$ and $\vert x- y \vert >0$
Background
There is a result in [SV02] that
Lemma 2.5.1 Assume that $\Gamma $ is a $C^1$ smooth Jordan curve. Suppose that $g(x,y)$ satifies
(1) Function $g(x,y)$ is continuous in the set $\mathbb{R}^...
- 269
2
votes
0
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135
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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 ...
- 662
3
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1
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166
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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 ...
- 438