Abstract: We establish a connection between generalized accretive operators introduced by F. E. Browder and the theory of quasisymmetric mappings in Banach spaces pioneered by J. Väisälä. The interplay of the two fields allows for geometric proofs of continuity, differentiability, and surjectivity of generalized accretive operators.
Abstract: We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.
Abstract: Continuing our investigation of quasiconformal mappings with convex potentials, we obtain a new characterization of quasiuniformly convex functions and improve our earlier results on the existence of quasiconformal mappings with prescribed sets of singularities.
Abstract: A hyperspace is a space of nonempty closed sets equipped with the Hausdorff metric. Among the subjects considered in this paper are Gromov hyperbolicity, quasisymmetric equivalence and bi-Lipschitz embeddings of hyperspaces.
Abstract: This paper concerns a class of monotone mappings in a Hilbert space that can be viewed as a nonlinear version of the class of positive invertible operators. Such mappings are proved to be open, locally Hölder continuous, and quasisymmetric. They arise naturally from the Beurling-Ahlfors extension and from Brenier's polar factorization, and find applications in the geometry of metric spaces and the theory of elliptic partial differential equations.
Abstract: We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson.
Abstract: This paper concerns convex functions that arise as potentials of quasiconformal mappings. Several equivalent definitions for such functions are given. We use them to construct quasiconformal mappings whose Jacobian determinants are singular on a prescribed set of Hausdorff dimension less than one.
Abstract: This paper is concerned with strong solutions of uniformly elliptic equations of non-divergence type in the plane. First, we use the notion of quasiregular gradient mappings to improve Morrey's theorem on the Hölder continuity of gradients of solutions. Then we show that the Gilbarg-Serrin equation does not produce the optimal Hölder exponent in the considered class of equations. Finally, we propose a conjecture for the best possible exponent and prove it under an additional restriction.
Abstract: Using rearrangements of matrix-valued sequences, we prove that with certain boundary conditions the solution of the one-dimensional Schrödinger equation increases or decreases under monotone rearrangements of its potential.
Abstract: We prove that quasiregular gradient mappings exhibit higher degree of Hölder continuity than the one that is optimal for general quasiregular mappings. This improves a classical result of Morrey on the regularity of strong solutions of uniformly elliptic PDEs with measurable coefficients. Our Hölder estimate for homogeneous solutions of such equations is close to the best possible.
Abstract: Using the normal family arguments we develop a fairly general method of constructing the G-limits of some differential operators.
Abstract: We prove that a K-quasiconformal mapping belongs to the little Hölder space c0,1/K if and only if its local modulus of continuity has an appropriate order of vanishing at every point. No such characterization is possible for Hölder spaces with exponent greater than 1/K.
Abstract: We study the behavior of a K-quasiregular mapping near points where its local modulus of continuity has order 1/K. We prove that the mapping is spherically analytic at such points and is asymptotically a rotation on circles. This result is used to prove sharp distortion estimates, including a version of Schwarz's lemma.
Abstract: We find a sufficient condition for a weakly differentiable homeomorphism in Euclidean space to have a homeomorphic extension to the boundary of its domain of definition. In a certain sense, this condition is best possible.
Abstract: We identify the equality cases in some of Dubinin's inequalities expressing the monotonicity of the generalized reduced modulus for open subsets of Cn and Rn. These results are applied to extremal problems of geometric function theory.
Abstract: A simple proof of the recent result by E. G. Emel'yanov concerning the maximum of the conformal radius r(D,1) for a family of simply connected domains with a fixed value r(D,0) is given. A similar problem is solved for a family of convex domains.
Abstract: The hyperbolic radius of a domain on the Riemann sphere is equal to the reciprocal of the density of the hyperbolic metric. In the present paper, it is proved that the hyperbolic radius is a convex function if and only if the complement of the domain is a convex set.
Abstract: The paper continues studies of reduced moduli of open sets. A notion of reduced modulus of the complex sphere is introduced, a formula for this modulus is obtained, and a number of its properties are proved. Applications of the reduced modulus of the sphere to various problems in geometric function theory are given.