1
vote
1answer
134 views
Uniqueness of classical solution with degenerate boundary
Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial d …
2
votes
0answers
74 views
Analytical continuation of electrostatic potentials
I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the …
4
votes
0answers
102 views
A finely open set, not open up to polar set?
I already asked this on M.SE, but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge …
0
votes
1answer
75 views
Higher dimensional analogue of Kellog’s theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)
Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the …
1
vote
1answer
413 views
Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?
I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \ …
10
votes
1answer
279 views
Modern version of an inequality of R. M. Gabriel for contour integrals
I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabrie …
3
votes
1answer
251 views
Books about Capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, whic …
4
votes
1answer
292 views
Logarithmic potential
Given a continuous, compactly supported function $f$ on $R^2$, it is known that the logarithmic potential of $f$, that is
$$
U_{f}(x):=-\frac{1}{2\pi}\int\log|x-y|f(y)dy
$$
has the …
0
votes
1answer
125 views
On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. T …
5
votes
2answers
448 views
Kähler potentials that depend only on geodesic distance
Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone …
1
vote
1answer
204 views
Boundary Value Problem in the space of Distributions
I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\part …
12
votes
2answers
899 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 i …
4
votes
2answers
359 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 D …
4
votes
0answers
292 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 t …
0
votes
1answer
145 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-t …

