Canonical mappings in
Catching sets with quasicircles.
We show how certain geometric conditions on a planar set imply that the set must lie on a quasicircle, and we give a geometric characterization of all subsets of the plane that are quasiconformally equivalent to the usual Cantor middle-third set.
Characteristic of quasispheres in terms of quasiconformal homogeneity.
Characterizations of quasidisks
Cluster sets and boundary behavior of quasiregular mappings.
Complex geometry of the universal Teichmüller space.
Conformal and quasiconformal mappings of close multiply-connected domains.
Conformal invariants in the punctured unit disk.
Constructing quasisymmetric functions via graph-directed iterated function systems.
Convex integration and the theory of elliptic equations
This paper deals with the theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions are based...
Convolution and Quasiconformal Extension
Counterexamples for unique continuation.
Covering of radial segments for domains bounded by -circles.