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1. CMB 2012 (vol 56 pp. 241)

Betsakos, Dimitrios; Pouliasis, Stamatis
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)

Osękowski, Adam
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)

Cegrell, Urban
Convergence in Capacity
In this note we study the convergence of sequences of Monge-Ampère measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

Keywords:complex Monge-Ampère operator, convergence in capacity, plurisubharmonic function
Categories:32U20, 31C15

4. CMB 2009 (vol 52 pp. 555)

Hirata, Kentaro
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)

Okoudjou, Kasso A.; Rogers, Luke G.; Strichartz, Robert S.
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)

Hanley, Mary
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)

Talvila, Erik
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)

Laugesen, Richard S.; Pritsker, Igor E.
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)

Miyamoto, Ikuko; Yanagishita, Minoru; Yoshida, Hidenobu
Beurling-Dahlberg-Sjögren 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)

Weitsman, Allen
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)

Bagby, Thomas; Gauthier, P. M.
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)

Khavinson, Dmitry
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

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