Expand all Collapse all | Results 1 - 12 of 12 |
1. CMB 2012 (vol 56 pp. 241)
Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius We prove variants of Schwarz's lemma involving monotonicity
properties of condenser capacity and inner radius. Also, we
examine when a similar monotonicity property holds for the
hyperbolic metric.
Keywords:condenser capacity, inner radius, hyperbolic metric, Schwarz's lemma Categories:30C80, 30F45, 31A15 |
2. CMB 2011 (vol 55 pp. 597)
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \|v\|_p \leq C_{p,\infty}
\|u\|_\infty,\quad
\|v\|_1 \leq C_{1,p} \|u\|_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.
Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem Categories:31B05, 60G44, 60G40 |
3. CMB 2011 (vol 55 pp. 242)
Convergence in Capacity In this note we study the convergence of sequences of Monge-AmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex Monge-AmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 |
4. CMB 2009 (vol 52 pp. 555)
Boundary Behavior of Solutions of the Helmholtz Equation This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewood-type theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 |
5. CMB 2009 (vol 52 pp. 105)
Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket We prove there exist exponentially decaying generalized eigenfunctions
on a blow-up of the Sierpinski gasket with boundary. These are used
to show a Borel-type theorem, specifically that for a prescribed jet
at the boundary point there is a smooth function having that jet.
Categories:28A80, 31C45 |
6. CMB 2008 (vol 51 pp. 229)
Existence of Solutions to Poisson's Equation Let $\Omega$ be a domain in $\mathbb R^n$ ($n\geq 2$). We find a
necessary and sufficient topological condition on $\Omega$ such
that, for any measure $\mu$ on $\mathbb R^n$, there is a function $u$
with specified boundary conditions that satisfies the Poisson
equation $\Delta u=\mu$ on $\Omega$ in the sense of distributions.
Category:31B25 |
7. CMB 2005 (vol 48 pp. 133)
Estimates of Henstock-Kurzweil Poisson Integrals If $f$ is a real-valued function on $[-\pi,\pi]$ that
is Henstock-Kurzweil integrable, let $u_r(\theta)$ be its Poisson
integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$
and this estimate is sharp for $1\leq p\leq\infty$.
If $\mu$ is a finite Borel measure and $u_r(\theta)$ is its Poisson
integral then for each $1\leq p\leq \infty$ the estimate
$\|u_r\|_p=O((1-r)^{1/p-1})$ as $r\to 1$ is sharp.
The Alexiewicz
norm estimates $\|u_r\|\leq\|f\|$ ($0\leq r<1$) and $\|u_r-f\|\to 0$
($r\to 1$) hold. These estimates lead to two uniqueness theorems for
the Dirichlet problem
in the unit disc with Henstock-Kurzweil integrable boundary data.
There are similar growth estimates when $u$ is in the harmonic Hardy
space associated with the Alexiewicz
norm and when $f$ is of bounded variation.
Categories:26A39, 31A20 |
8. CMB 2003 (vol 46 pp. 373)
Potential Theory of the Farthest-Point Distance Function We study the farthest-point distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$-gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 |
9. CMB 2003 (vol 46 pp. 252)
Beurling-Dahlberg-SjÃ¶gren Type Theorems for Minimally Thin Sets in a Cone This paper shows that some characterizations of minimally thin sets
connected with a domain having smooth boundary and a half-space in
particular also hold for the minimally thin sets at a corner point of
a special domain with corners, {\it i.e.}, the minimally thin set at
$\infty$ of a cone.
Categories:31B05, 31B20 |
10. CMB 2002 (vol 45 pp. 154)
On the Poisson Integral of Step Functions and Minimal Surfaces Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 |
11. CMB 1998 (vol 41 pp. 257)
Note on the support of Sobolev functions We prove a topological restriction on the support of Sobolev functions.
Categories:46E35, 31B05 |
12. CMB 1997 (vol 40 pp. 60)
Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potential-theoretic proof of a theorem of
G.~Johnsson: all solutions of Cauchy's problems for the Laplace
equations with an entire data on a sphere extend harmonically to
the whole space ${\bf R}^N$ except, perhaps, for the center of the
sphere.
Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics Categories:35B60, 31B20 |