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7
votes
2answers
508 views

Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator. We wish to solve the Dirichlet problem ...
3
votes
0answers
266 views

Best Poincare constants on the surface of a ball

I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first ...
0
votes
1answer
157 views

Properties of the quadruple layer potential

What is known about the quadruple layer potential in 3D (on closed smooth surfaces)? In terms of jump relations, continuity on Hölder Spaces (and/or Sobolev spaces), and Calderon-type identities ...
8
votes
1answer
545 views

How does electric potential relate to mean curvature?

Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$. Now consider the electric potential generated by this uniform mass ...
12
votes
2answers
1k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
2
votes
1answer
490 views

Compact sets of the complex plane having the K-property ?

I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples. As it is ...
4
votes
0answers
331 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...
4
votes
0answers
865 views

A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions. First recall the Cauchy integral formula: Let $U$ be an open subset of ...
5
votes
2answers
5k views

Version of the Poincaré Inequality

Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality $$\|u\|_p \le C \|\nabla u\|_p$$ ($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used ...