Interpolating between torsional rigidity and principal frequency

A one-parameter family of variational problems is introduced that interpolates between torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. The associated partial differential equation is derived, which is shown to have positive solutions in many cases. Results are obtained regarding extremal domains and regarding variations of the domain or the parameter.Comment: 13 pages, 2 figure

Sharp Integrability for Brownian Motion in Parabola-shaped Regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains {\cal P}_\alpha = \{(x,y)\in \bR\times \bR\colon x> 0, |y| < Ax^{\alpha}\}, $0<\alpha<1$. Together with some simple good-$\lambda$ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for $\alpha =1/2$ by Ba\~nuelos, DeBlassie and Smits \cite{ba} and for general $\alpha$ by Li \cite{li}. A sharp version of this result is also proved in higher dimensions

The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing

By modifying a domain first suggested by Ruth Goodman in 1935 and by exploiting the explicit solution by Fedorov of the Poly\'a-Chebotarev problem in the case of four symmetrically placed points, an improved upper bound for the univalent Bloch-Landau constant is obtained. The domain that leads to this improved bound takes the form of a disk from which some arcs are removed in such a way that the resulting simply connected domain is harmonically symmetric in each arc with respect to the origin. The existence of domains of this type is established, using techniques from conformal welding, and some general properties of harmonically symmetric arcs in this setting are established

Configurations of balls in Euclidean space that Brownian motion cannot avoid

We consider a collection of balls in Euclidean space and the problem of determining if Brownian motion has a positive probability of avoiding all the ball