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

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

A hunt for sharp LpL ^p-estimates and rank-one convex variational integrals

Learning how to figure out sharp $L^p$-estimates of nonlinear differential expressions, to prove and use them, is a fundamental part of the development of PDEs and Geometric Function Theory (GFT). Our survey presents, among what is known to date, some notable recent efforts and novelties made in this direction. We focus attention here on the historic Morrey's Conjecture and Burkholder's martingale inequalities for stochastic integrals. Some of these topics have already been discussed by the present authors [5] and by Rodrigo Ba\~{n}uelos [10]. Nevertheless, there is always something new to add.Comment: 24 page