Homeomorphic Solutions to Reduced Beltrami Equations

We study differential expressions related to linear families of quasiconformal mappings and give a simple and direct proof to a result due to Alessandrini and Nesi arXiv:0707.0727.Comment: 8 pages, a typo corrected (page 4, line 16), references added, and the text flow improve

Asymptotic variance of the Beurling transform

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $\Sigma^2 = 1$. In this paper, we prove $0.87913 \le \Sigma^2 \le 1$. For the lower bound, we give examples of polynomial Julia sets which are $k$-quasicircles with dimensions $1+ 0.87913 \, k^2$ for $k$ small, thereby showing that $\Sigma^2 \ge 0.87913$. The key ingredient in this construction is a good estimate for the distortion $k$, which is better than the one given by a straightforward use of the $\lambda$-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $\Sigma^2$ in terms of nearly circular polynomial Julia sets.Comment: 45 page

Nonlinear Fourier analysis for discontinuous conductivities: computational results

Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calder\'on problem exposed in [Nachman; Annals of Mathematics 143, 1996] and [Astala and P\"aiv\"arinta; Annals of Mathematics 163, 2006]: the Astala-P\"aiv\"arinta theory-based "low-pass transport matrix method" implemented in [Astala et al.; Inverse Problems and Imaging 5, 2011] and the "shortcut method" which considers ingredients of both theories. The latter method is formally similar to the Nachman theory-based regularized EIT reconstruction algorithm studied in [Knudsen, Lassas, Mueller and Siltanen; Inverse Problems and Imaging 3, 2009] and several references from there. New numerical results are presented using parallel computation with size parameters larger than ever, leading mainly to two conclusions as follows. First, both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases, and there seems to be a Gibbs-like phenomenon producing ringing artifacts. Second, the transport matrix method loses accuracy away from a (freely chosen) pivot point located outside of the object to be studied, whereas the shortcut method produces reconstructions with more uniform quality.Comment: 29 page

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We give sufficient conditions for quasiconformal mappings between simply connected Lipschitz domains to have H\"older, Sobolev and Triebel-Lizorkin regularity in terms of the regularity of the boundary of the domains and the regularity of the Beltrami coefficients of the mappings. The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in the context of quasiconformal mappings.Comment: 45 pages, 3 figure

Random Curves by Conformal Welding

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.Comment: 5 page