Mappings with convex potentials and the quasiconformal Jacobian problem

Conformal dimension does not assume values between zero and one

“…Sections 2 and 3 contain preliminaries concerning nonlinear accretive operators and Lebesgue-Bochner spaces. Some auxiliary results are reminiscent of [16]. A part of Lemma 2.6 was inspired by [14].…”

Quasiconformal geometry of monotone mappings

“…Convex functions with round sections are intimately connected to quasisymmetric and δ-monotone mappings. These connections are summarized in the following theorem, which builds on the results of [24]. (v) u has round sections.…”

“…Finally, (ii) is a 2-local property because it is equivalent to (i) by virtue of Theorem 6 and Lemma 18. Theorem 3.1 in [24] asserts the quantitative equivalence of (ii), (iii), (iv) and (v) for convex functions in R d (here, we only need the case d = 2). By virtue of 2-locality, this proves the quantitative equivalence of (i), (ii), (iii), (iv) and (v) in an arbitrary Hilbert space.…”

“…It is proved that the Beurling-Ahlfors extensions [6] of some quasisymmetric functions are not only quasiconformal, but also δmonotone. Section 4 builds on an earlier paper [24]. Its main result is that for gradients of convex functions the notions of quasisymmetry and δ-monotonicity coincide.…”

“…This result supports the above claim that the class of δ-monotone mappings is a natural counterpart of the bi-Lipschitz class. The final part of the paper relates δ-monotone mappings to the quasiconformal Jacobian problem [8,24] and to strong solutions of elliptic equations [2][3][4].…”